\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1.05in]{geometry} \usepackage{booktabs,microtype,array} \makeatletter \newcommand{\needspace}[1]{\par\penalty-100\begingroup\setlength{\dimen@}{#1}\dimen@ii\pagegoal \advance\dimen@ii-\pagetotal\ifdim\dimen@>\dimen@ii\newpage\fi\endgroup} \makeatother \usepackage{lmodern,textcomp} \usepackage{listings,xcolor} \usepackage[hidelinks]{hyperref} \hypersetup{pdftitle={Strongly regular graphs on 160, 232, 261 and 280 vertices from fusions of orbitals of PGammaL(2,8) and 2**4:(S3xS3)},pdfauthor={Anton Koval'}} \lstset{basicstyle=\ttfamily\scriptsize,breaklines=true,columns=fullflexible,keepspaces=true, showstringspaces=false,numbers=left,numberstyle=\tiny\color{gray},frame=none,language=Python, commentstyle=\color{gray!60!black},keywordstyle=\bfseries,upquote=false} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{construction}[theorem]{Construction} \newtheorem{computation}[theorem]{Computation} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \newcommand{\F}{\mathbb{F}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\srg}{\mathrm{srg}} \newcommand{\Aut}{\mathrm{Aut}} \newcommand{\PG}{\mathrm{PG}} \newcommand{\PGL}{\mathrm{PGL}} \newcommand{\PGaL}{\mathrm{P\Gamma L}} \newcommand{\SL}{\mathrm{SL}} \newcommand{\PSL}{\mathrm{PSL}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\code}[1]{\texttt{#1}} \newcommand{\brk}{\allowbreak{}} \newcommand{\Pone}{\mathcal P} \newcommand{\Ptwo}{\mathcal L} \newcommand{\Eset}{\mathcal E} \emergencystretch=2em \title[Strongly regular graphs on 160, 232, 261 and 280 vertices]{\texorpdfstring{Strongly regular graphs on $160$, $232$, $261$ and $280$ vertices from fusions of orbitals of $\PGaL(2,8)$ and $2^4{:}(S_3\times S_3)$}{Strongly regular graphs on 160, 232, 261 and 280 vertices}} \author{Anton Kova\v{l}} \address{AI Lead, V\'UB banka, Bratislava, Slovakia\newline ORCID 0009-0009-3182-6626} \email{akoval@pobox.sk} \date{October 2026. Draft, version 3. Code and data: see the Data availability statement.} \begin{document} \begin{abstract} We construct strongly regular graphs with parameters $(160,54,18,18)$, $(232,63,14,18)$, $(261,64,14,16)$ and $(280,63,14,14)$, and hence also their complements. All four parameter sets were marked ``?'' (existence unknown) in Brouwer's online table on 11 and 12 October 2026; they are the parameters of the hypothetical partial geometries $\mathrm{pg}(9,5,3)$, $\mathrm{pg}(7,8,2)$, $\mathrm{pg}(8,7,2)$ and $\mathrm{pg}(9,6,2)$. We give eight pairwise non-isomorphic graphs: three on $160$ vertices, one on $232$, two on $261$ and two on $280$. None of them is geometric. As far as we can determine, they are the first graphs known with these parameters. Five of the graphs have a short description over the projective line $\PG(1,8)$ and its quadratic extension $\PG(1,64)$. The vertices are pairs of points of $\PG(1,8)$ or conjugate pairs of points of $\PG(1,64)\setminus\PG(1,8)$, each carrying a label in $\F_8^*$, together with a small orbit of points, pairs, or conjugate pairs. Adjacency is decided by cross ratios: for two labelled vertices on disjoint pairs, the quotient (or the product) of the two labels must equal the cross ratio of the two pairs or its inverse; for pairs sharing a point it must equal $1$, and a labelled pair and a conjugate pair are adjacent when their cross ratio is a primitive cube root of unity. The ``quotient'' graphs on $232$, $261$ and $280$ vertices have the automorphism group $(\SL(2,8)\times C_7)\rtimes C_6\cong\GL(2,8)\rtimes C_6$ of order $21168$; the two ``product'' graphs have $C_2\times\PGaL(2,8)$, of order $3024$. One graph on $160$ vertices lives on the $16$ cells of a $4\times4$ grid and three copies of the $48$ flags of its even transversals; its automorphism group is $C_3\times2^4{:}(S_3\times S_3)$, of order $1728$. The other two graphs on $160$ vertices have automorphism groups of orders $1728$ and $576$. The graphs were found by a computer search over fusions of orbitals of groups acting with two or three orbits, using the GAP library of tables of marks; the search reported five graphs, and enumerating all fusions of the same group actions gave three more. We do not classify the graphs with these parameters. The same constructions over $\PG(1,q)$ give, for $q=4$, the classical skew-lines graph of $\PG(3,2)$ and a second known $\srg(35,16,6,8)$, and no strongly regular graph for $q=16$ or $q=32$ under the groups searched (complete solver verdicts): within this model the graphs for $q=8$ appear to be sporadic. Strong regularity of every graph is checked by an exact, standalone, standard-library Python program, the constructions are re-implemented from the text of this paper, and all computer results were reproduced by an independent audit with separately written code and checked again by a referee with its own code. \end{abstract} \maketitle \section{Introduction} A \emph{strongly regular graph} $\srg(v,k,\lambda,\mu)$ is a $k$-regular graph on $v$ vertices, neither complete nor edgeless, in which two adjacent vertices have exactly $\lambda$ common neighbours and two distinct non-adjacent vertices exactly $\mu$. Equivalently, its adjacency matrix $A$ satisfies $A^2=kI+\lambda A+\mu(J-I-A)$. Brouwer maintains a table of feasible parameter sets with $v\le1300$ and their status \cite{BrouwerTable}; the book of Brouwer and Van Maldeghem \cite{BvM} contains the same table as of its publication. A partial geometry $\mathrm{pg}(s,t,\alpha)$ has lines of size $s+1$, $t+1$ lines on every point, and for every non-incident point--line pair exactly $\alpha$ points of the line are collinear with the point. Its collinearity graph is an $\srg\bigl((s+1)(st+\alpha)/\alpha,\ s(t+1),\ s-1+t(\alpha-1),\ \alpha(t+1)\bigr)$, and a strongly regular graph with these parameters is called \emph{pseudo-geometric}. It is \emph{geometric} if it is the collinearity graph of such a geometry \cite[Ch.~1]{BvM}. The four parameter sets of this paper are pseudo-geometric (Table~\ref{tab:params}). For $(160,54,18,18)$ the table records that $\mathrm{pg}(9,5,3)$ does not exist, so a graph with these parameters cannot be geometric. For the other three sets the table asks whether the partial geometry exists. The table recorded no graph for any of the four sets. \begin{table}[ht] \caption{The parameter sets. ``Comment'' is the comment in \cite{BrouwerTable} as read on 12 October 2026, where each row was marked ``?''. The eigenvalues follow from the parameters. ``Del.'' is the Delsarte clique bound $1-k/s$ and ``Hoff.'' the Hoffman coclique bound $v(-s)/(k-s)$.}\label{tab:params} \centering\small \begin{tabular}{@{}lllrr>{\raggedright\arraybackslash}p{3.6cm}@{}} \toprule parameters & spectrum & complement & Del. & Hoff. & comment in \cite{BrouwerTable}\\ \midrule $(160,54,18,18)$ & $54^1\,6^{75}\,(-6)^{84}$ & $(160,105,68,70)$ & $10$ & $16$ & pg(9,5,3) does not exist (no 2-graph$^*$ for line graph)\\ $(232,63,14,18)$ & $63^1\,5^{144}\,(-9)^{87}$ & $(232,168,122,120)$ & $8$ & $29$ & pg(7,8,2)?\\ $(261,64,14,16)$ & $64^1\,6^{144}\,(-8)^{116}$ & $(261,196,147,147)$ & $9$ & $29$ & pg(8,7,2)?\\ $(280,63,14,14)$ & $63^1\,7^{135}\,(-7)^{144}$ & $(280,216,166,168)$ & $10$ & $28$ & pg(9,6,2)?\\ \bottomrule \end{tabular} \end{table} \needspace{10\baselineskip} \begin{theorem}\label{thm:main}\leavevmode \begin{enumerate} \item There exist strongly regular graphs with parameters $(160,54,18,18)$, $(232,63,14,18)$, $(261,64,14,16)$ and $(280,63,14,14)$. On these parameter sets there are at least $3$, $1$, $2$ and $2$ pairwise non-isomorphic graphs, listed in Table~\ref{tab:graphs}. \item The graphs $\Gamma_{232}$, $\Gamma_{261}^{Q}$, $\Gamma_{261}^{P}$, $\Gamma_{280}^{Q}$, $\Gamma_{280}^{P}$ of Construction~\ref{con:pg18} are defined over $\PG(1,8)\subset\PG(1,64)$ by cross ratios. The group $\Gamma_8=(\SL(2,8)\times C_7)\rtimes(C_3\times C_2)$ of order $21168$ acts on the three ``quotient'' graphs $\Gamma^Q$ and is their full automorphism group; $C_2\times\PGaL(2,8)$ acts on the two ``product'' graphs $\Gamma^P$ and is their full automorphism group. \item The graph $\Gamma_{160}^{(1)}$ of Construction~\ref{con:160} lives on the cells of a $4\times4$ grid and three copies of the flags of its even transversals; $\Aut\Gamma_{160}^{(1)}\cong C_3\times2^4{:}(S_3\times S_3)$, of order $1728$. The graphs $\Gamma_{160}^{(2)}$ and $\Gamma_{160}^{(3)}$ are fusions of orbitals of $2^4{:}(S_3\times S_3)$ acting on $96+48+16$ points, with automorphism groups of orders $1728$ and $576$. \item None of the eight graphs is geometric. $\Gamma_{160}^{(i)}$, $\Gamma_{280}^{Q}$ and $\Gamma_{280}^P$ have no Delsarte clique at all; $\Gamma_{261}^P$ and $\Gamma_{261}^Q$ have only $46$ resp.\ $100$ Delsarte cliques, fewer than the $232$ lines of a $\mathrm{pg}(8,7,2)$; and no subfamily of the $513$ Delsarte cliques of $\Gamma_{232}$ partitions its edge set. \end{enumerate} \end{theorem} \emph{What is proved by hand, and what by computer.} The constructions, the valencies, the invariance of each graph under the named group (hence the lower bounds on $|\Aut|$), the orbit structure of that group, the local graph of a point of $\Gamma_{261}^Q$ and several cliques and cocliques are proved by hand (Section~\ref{sec:con}). We have not written a complete hand proof of the values of $\lambda$ and $\mu$; strong regularity of every graph is established by exact computation, by several programs written independently of each other (Section~\ref{sec:verify}), among them one written from the text of Section~\ref{sec:con} alone. The orders of the automorphism groups, the non-isomorphism, the clique and coclique counts behind ``not geometric'' and the family results of Section~\ref{sec:family} are computer results. Table~\ref{tab:status} lists every claim. \emph{Scope.} Every completeness statement in this paper concerns all unions of orbitals of one stated group action (``all fusions of one action''). We make no classification claim for any of the four parameter sets: there may be further graphs with these parameters, with other groups or with no symmetry at all. \emph{Relation to earlier work.} In earlier papers we found strongly regular graphs as unions of orbitals of transitive permutation groups \cite{KovalZ} and of groups acting with two orbits \cite{KovalMO}. The present graphs come from the next step of the same programme: the search of \cite{KovalMO} extended to three orbits and to direct products $G\times C_m$ (Section~\ref{sec:search}). None of the eight graphs is vertex-transitive: each automorphism group has at least two orbits on vertices (Table~\ref{tab:graphs}). So no search over transitive actions could have found them. \subsection*{Conventions} An $\srg(v,k,\lambda,\mu)$ has the eigenvalues $k>r>s$ with $r,s=\frac12\bigl(\lambda-\mu\pm\sqrt{(\lambda-\mu)^2+4(k-\mu)}\bigr)$. A clique $C$ has $|C|\le1-k/s$ (Delsarte bound); a clique attaining it is a \emph{Delsarte clique}, and every vertex outside it has exactly $|C|(k-|C|+1)/(v-|C|)$ neighbours in it. A coclique $S$ has $|S|\le v(-s)/(k-s)$ (Hoffman bound); a coclique attaining it is a \emph{Hoffman coclique}, and every vertex outside it has exactly $k|S|/(v-|S|)$ neighbours in it \cite[Ch.~1]{BvM}. If $\Gamma$ is the collinearity graph of a partial geometry, its lines are cliques whose size equals the Delsarte bound of $\Gamma$, i.e.\ Delsarte cliques, and they partition the edge set; so a pseudo-geometric graph whose Delsarte cliques do not contain a partition of the edge set is not geometric. Groups are named as in the ATLAS and as in GAP's \code{StructureDescription}; ``$:$'' is a split extension. TomLib class numbers refer to the GAP package TomLib \cite{TomLib} version 1.2.11 with GAP 4.12.1 \cite{GAP}. Notation: $\Pone$, $\Ptwo$, $\Eset$ are the sets of points, pairs and conjugate pairs of Section~\ref{sec:cr}; $\Gamma_q$ ($q$ a power of $2$) denotes a group (Proposition~\ref{prop:inv} and Section~\ref{sec:family}), while $\Gamma_{232}$, $\Gamma^\star_{261}$, $\Gamma^\star_{280}$ and $\Gamma^{(i)}_{160}$, indexed by the number of vertices, denote graphs; $W$ denotes the group $2^4{:}(S_3\times S_3)$ of Section~\ref{sec:160}. \begin{table}[ht] \caption{The eight graphs ($\Pone$, $\Ptwo$, $\Eset$: notation of Section~\ref{sec:cr}). ``No.'' is the label used in the data files (Data availability). ``Orbits'' are the orbits of $\Aut$ on vertices. All entries are computer results except the constructions; Section~\ref{sec:con} proves $|\Aut|\ge$ the order listed for $\Gamma_{232}$, $\Gamma^{P}$, $\Gamma^{Q}$ and $\Gamma^{(1)}_{160}$ by hand.}\label{tab:graphs} \centering\small \begin{tabular}{@{}lllrll@{}} \toprule No. & graph & construction & $|\Aut|$ & $\Aut$ & orbits\\ \midrule G160 & $\Gamma^{(1)}_{160}$ & grid and flags (Constr.~\ref{con:160}) & $1728$ & $C_3\times2^4{:}(S_3\times S_3)$ & $144+16$\\ G160b & $\Gamma^{(2)}_{160}$ & fusion, $2^4{:}(S_3\times S_3)$ on $96+48+16$ & $1728$ & $C_3{:}(2^4{:}(S_3\times S_3))$ & $144+16$\\ G160c & $\Gamma^{(3)}_{160}$ & fusion, $2^4{:}(S_3\times S_3)$ on $96+48+16$ & $576$ & $2^4{:}(S_3\times S_3)$ & $96+48+16$\\ G232 & $\Gamma_{232}$ & quotient rule on $\Eset\times\F_8^*\cup\Ptwo$ & $21168$ & $\GL(2,8)\rtimes C_6$ & $196+36$\\ G261a & $\Gamma^{P}_{261}$ & product rule on $\Ptwo\times\F_8^*\cup\Pone$ & $3024$ & $C_2\times\PGaL(2,8)$ & $216+36+9$\\ G261b & $\Gamma^{Q}_{261}$ & quotient rule on $\Ptwo\times\F_8^*\cup\Pone$ & $21168$ & $\GL(2,8)\rtimes C_6$ & $252+9$\\ G280a & $\Gamma^{P}_{280}$ & product rule on $\Ptwo\times\F_8^*\cup\Eset$ & $3024$ & $C_2\times\PGaL(2,8)$ & $216+36+28$\\ G280b & $\Gamma^{Q}_{280}$ & quotient rule on $\Ptwo\times\F_8^*\cup\Eset$ & $21168$ & $\GL(2,8)\rtimes C_6$ & $252+28$\\ \bottomrule \end{tabular} \end{table} \section{The constructions}\label{sec:con} \subsection{\texorpdfstring{Cross ratios over $\F_8\subset\F_{64}$}{Cross ratios over F8 in F64}}\label{sec:cr} Let $\F_{64}\supset\F_8$, and write $\sigma\colon x\mapsto x^8$ for the involutory automorphism of $\F_{64}$ over $\F_8$. Points of $\PG(1,64)$ are written $\langle u\rangle$ with $u\in\F_{64}^2\setminus\{0\}$; for $u=(u_1,u_2)$ and $w=(w_1,w_2)$ put $[u,w]=u_1w_2+u_2w_1$ (the determinant; signs do not matter in characteristic $2$, so $[u,w]=[w,u]$). We use three sets of objects: \begin{itemize} \item $\Pone=\PG(1,8)$, the $9$ points of the subline; \item $\Ptwo$, the $36$ unordered pairs of distinct points of $\Pone$; \item $\Eset$, the $28$ pairs $\{e,e^\sigma\}$ with $e\in\PG(1,64)\setminus\PG(1,8)$ (these are the $56$ points off the subline, paired by $\sigma$). \end{itemize} Both $\Ptwo$ and $\Eset$ consist of $2$-subsets of $\PG(1,64)$, and any two distinct members of $\Ptwo\cup\Eset$ are disjoint unless both lie in $\Ptwo$ and share a point. For disjoint $2$-subsets $X=\{a,b\}$ and $Y=\{c,d\}$ put \[ \chi(X,Y)=\frac{[a,c]\,[b,d]}{[a,d]\,[b,c]}, \] computed with any representatives; the value does not depend on them, since numerator and denominator are homogeneous of the same degree in each of $a,b,c,d$. \begin{lemma}\label{lem:cr} Let $X,Y$ be disjoint $2$-subsets of $\PG(1,64)$. \begin{enumerate} \item Swapping $a$ and $b$, or $c$ and $d$, replaces $\chi$ by $\chi^{-1}$, and $\chi(Y,X)=\chi(X,Y)$. So the unordered pair $\{\chi(X,Y),\chi(X,Y)^{-1}\}$ depends only on $\{X,Y\}$. \item $\chi(X,Y)\notin\{0,1\}$. \item If $X,Y\in\Ptwo$ or $X,Y\in\Eset$, then $\chi(X,Y)\in\F_8^*\setminus\{1\}$. If $X\in\Ptwo$ and $Y\in\Eset$, then $\chi(X,Y)^9=1$. \item $\chi(gX,gY)=\chi(X,Y)$ for every $g\in\GL(2,64)$, and $\chi(X^\phi,Y^\phi)=\chi(X,Y)^2$ for the Frobenius map $\phi\colon(u_1,u_2)\mapsto(u_1^2,u_2^2)$. \end{enumerate} \end{lemma} \begin{proof} (1) is immediate from $[u,w]=[w,u]$. (2) The denominator and numerator are non-zero because the four points are distinct. The Pl\"ucker relation $[a,b][c,d]+[a,c][b,d]+[a,d][b,c]=0$ (valid in characteristic $2$ without signs) shows $[a,c][b,d]=[a,d][b,c]$ iff $[a,b][c,d]=0$, which is false. (3) Apply $\sigma$ coordinatewise. Points of $\Pone$ are fixed and $\sigma$ swaps the two points of a member of $\Eset$. For $X,Y\in\Ptwo$ nothing moves, for $X,Y\in\Eset$ both pairs are swapped, so in both cases $\chi^\sigma=\chi$ by (1), i.e.\ $\chi\in\F_8$. For $X\in\Ptwo$, $Y\in\Eset$ only $c,d$ are swapped, so $\chi^8=\chi^{-1}$. (4) $[gu,gw]=\det(g)[u,w]$ and $\det g$ cancels; $\phi$ is a field automorphism applied to all coordinates. \end{proof} Fix a generator $\omega$ of $\F_8^*$ and the two primitive cube roots of unity $\zeta,\zeta^2\in\F_4\subset\F_{64}$. Since $3\mid 9$, they are among the values allowed for $\chi(X,Y)$ with $X\in\Ptwo$, $Y\in\Eset$ by Lemma~\ref{lem:cr}(3). \begin{lemma}\label{lem:counts}\leavevmode \begin{enumerate} \item For $\pi\in\Ptwo$ and $x\in\F_8^*\setminus\{1\}$, exactly $7$ pairs $\pi'\in\Ptwo$ disjoint from $\pi$ have $\chi(\pi,\pi')\in\{x,x^{-1}\}$. \item For $\pi\in\Ptwo$, exactly $7$ elements $Y\in\Eset$ have $\chi(\pi,Y)\in\{\zeta,\zeta^2\}$; for $Y\in\Eset$, exactly $9$ pairs $\pi\in\Ptwo$ have this property. \end{enumerate} \end{lemma} \begin{proof} $\SL(2,8)$ is $2$-transitive on $\Pone$ and preserves $\chi$, so we may take $\pi=\{0,\infty\}$ with $0=\langle(0,1)\rangle$, $\infty=\langle(1,0)\rangle$. For a point $\langle(a,1)\rangle$ we have $[0,a]=a$ and $[\infty,a]=1$. (1) For $\pi'=\{a,b\}\subset\F_8^*$, $\chi(\pi,\pi')=a/b$, and the unordered pairs $\{a,b\}$ with $\{a/b,b/a\}=\{x,x^{-1}\}$ are the $7$ pairs $\{a,a/x\}$ (note $x\ne x^{-1}$). (2) For $Y=\{\langle(\varepsilon,1)\rangle,\langle(\varepsilon^8,1)\rangle\}$, $\chi(\pi,Y)=\varepsilon/\varepsilon^8=\varepsilon^{-7}$. This is a primitive cube root of unity iff the order of $\varepsilon$ divides $21$ but not $7$. There are $21-7=14$ such $\varepsilon$, none in $\F_8$, closed under $\sigma$; they form $7$ elements of $\Eset$. The number of such $\pi$ is the same for every $Y\in\Eset$, since $\SL(2,8)$ is transitive on $\Eset$ (Proposition~\ref{prop:inv}(3), whose proof does not use this lemma) and preserves $\chi$; double counting, $36\cdot7=28\cdot9$, gives $9$. \end{proof} \subsection{\texorpdfstring{The graphs on $232$, $261$ and $280$ vertices}{The graphs on 232, 261 and 280 vertices}}\label{sec:pg18} Vertices of the form $(X,c)$ carry a \emph{label} $c\in\F_8^*$. For two labelled vertices $(X,c)$, $(X',c')$ the \emph{quotient rule} uses $d=c'/c$ and the \emph{product rule} uses $d=c'c$; in both cases $d$ is compared with $\chi(X,X')^{\pm1}$, so by Lemma~\ref{lem:cr}(1) the conditions below are symmetric in the two vertices (for the quotient rule, swapping the vertices replaces $d$ by $d^{-1}$). \begin{construction}\label{con:pg18} Let $\star\in\{Q,P\}$ select the quotient rule ($Q$) or the product rule ($P$). \begin{enumerate} \item $\Gamma^\star_{261}$ has vertex set $(\Ptwo\times\F_8^*)\cup\Pone$, $252+9$ vertices. \begin{itemize} \item $(\pi,c)\sim(\pi,c')$ for $c\neq c'$: the $7$ labelled copies of a pair form a clique. \item For $|\pi\cap\pi'|=1$: $(\pi,c)\sim(\pi',c')$ iff $d=1$. \item For $\pi\cap\pi'=\emptyset$: $(\pi,c)\sim(\pi',c')$ iff $d\in\{\chi(\pi,\pi'),\chi(\pi,\pi')^{-1}\}$. \item $(\pi,c)\sim p$ iff $p\in\pi$; the $9$ points form a clique. \end{itemize} \item $\Gamma^\star_{280}$ has vertex set $(\Ptwo\times\F_8^*)\cup\Eset$, $252+28$ vertices. \begin{itemize} \item Two copies of the same pair are not adjacent. \item For $|\pi\cap\pi'|=1$ and for $\pi\cap\pi'=\emptyset$: as in (1). \item $(\pi,c)\sim Y$ iff $\chi(\pi,Y)\in\{\zeta,\zeta^2\}$; $\Eset$ is a coclique. \end{itemize} \item $\Gamma_{232}=\Gamma^Q_{232}$ has vertex set $(\Eset\times\F_8^*)\cup\Ptwo$, $196+36$ vertices. \begin{itemize} \item $(Y,c)\sim(Y',c')$ iff $Y\neq Y'$ and $c'/c\in\{\chi(Y,Y'),\chi(Y,Y')^{-1}\}$. \item $(Y,c)\sim\pi$ iff $\chi(\pi,Y)\in\{\zeta,\zeta^2\}$. \item $\pi\sim\pi'$ iff $|\pi\cap\pi'|=1$; on $\Ptwo$ this is the triangular graph $T(9)$. \end{itemize} \end{enumerate} \end{construction} The product rule in (3) does not give a strongly regular graph (it has two values of $\lambda$ and of $\mu$; Section~\ref{sec:search}). The choice of $\omega$, $\zeta$ and of the model of $\F_{64}$ does not matter: any two choices differ by a field isomorphism, which maps one graph onto the other. Replacing $\chi^{\pm1}$ by $\chi^{\pm2}$ in the rules gives isomorphic graphs: the map $(X,c)\mapsto(X,c^2)$ squares $d$ and leaves $\chi$ unchanged, so it maps the graph with exponent $\pm1$ onto the graph with exponent $\pm2$, and likewise for every exponent $\pm m$ with $7\nmid m$. The exponent $2$ was also checked by computer. Section~\ref{sec:hits} shows that the exponents $\pm1$, $\pm2$, $\pm3$ account exactly for the multiplicities of the fusions found. \begin{proposition}\label{prop:inv}\leavevmode \begin{enumerate} \item $\Gamma^\star_{261}$ is $64$-regular, and $\Gamma^\star_{280}$ and $\Gamma_{232}$ are $63$-regular. \item Let $\SL(2,8)$ act on points (and so on $\Ptwo$, $\Eset$), $t$ on labels by $c\mapsto\omega c$, $\phi$ by the Frobenius map on coordinates and $c\mapsto c^2$, and $\iota$ by $c\mapsto c^{-1}$. These generate a group $\Gamma_8=(\SL(2,8)\times\langle t\rangle)\rtimes(\langle\phi\rangle\times\langle\iota\rangle)$ of order $504\cdot7\cdot3\cdot2=21168$, which acts faithfully on each vertex set and preserves the graphs $\Gamma^Q_{261}$, $\Gamma^Q_{280}$ and $\Gamma_{232}$. Its subgroup $\langle\SL(2,8),\phi,\iota\rangle\cong C_2\times\PGaL(2,8)$, of order $3024$, preserves $\Gamma^P_{261}$ and $\Gamma^P_{280}$. \item The orbits of $\Gamma_8$ are $\Ptwo\times\F_8^*$ and $\Pone$, resp.\ $\Eset$, resp.\ $\Eset\times\F_8^*$ and $\Ptwo$. The orbits of $C_2\times\PGaL(2,8)$ on $\Ptwo\times\F_8^*$ are $\Ptwo\times\{1\}$ ($36$) and $\Ptwo\times(\F_8^*\setminus\{1\})$ ($216$). \end{enumerate} \end{proposition} \begin{proof} (1) In $\Gamma^\star_{261}$ a vertex $(\pi,c)$ has $6$ neighbours among the copies of $\pi$, one neighbour among the copies of each of the $14$ pairs meeting $\pi$ in a point, two among the copies of each of the $21$ pairs disjoint from $\pi$ (Lemma~\ref{lem:cr}(2): $\chi\ne\chi^{-1}$), and the $2$ points of $\pi$: $6+14+42+2=64$. A point has $8$ point neighbours and the $8\cdot7=56$ copies of the pairs through it. In $\Gamma^\star_{280}$ a vertex $(\pi,c)$ has $14+42+7=63$ neighbours (Lemma~\ref{lem:counts}(2)), and $Y\in\Eset$ has the $7$ copies of each of $9$ pairs. In $\Gamma_{232}$, $(Y,c)$ has two neighbours among the copies of each of the other $27$ elements of $\Eset$ and $9$ neighbours in $\Ptwo$; a pair has $14$ neighbours in $T(9)$ and $7\cdot7=49$ in $\Eset\times\F_8^*$. (2) The adjacency conditions involve only intersection sizes, the cross ratio, membership in $\{\zeta,\zeta^2\}$ and $d$. $\SL(2,8)$ fixes labels and preserves $\chi$ (Lemma~\ref{lem:cr}(4)); it commutes with $\sigma$, so it permutes $\Eset$. $t$ fixes $d=c'/c$. $\phi$ squares $\chi$ and $d$, and preserves $\{\zeta,\zeta^2\}$; it commutes with $\sigma$, so it permutes $\Eset$. $\iota$ inverts $d$ under both rules. So $\Gamma_8$ preserves the quotient graphs, and $\langle\SL(2,8),\phi,\iota\rangle$ preserves the product graphs ($t$ multiplies $d=cc'$ by $\omega^2$ and is not used). \emph{$\phi$ has order $3$ on the vertices.} $\phi^3$ is $x\mapsto x^8$ on coordinates, i.e.\ $\sigma$; it fixes every point of $\Pone$ (hence every member of $\Ptwo$), maps each member $\{e,e^\sigma\}$ of $\Eset$ to itself, and fixes every label ($c^8=c$ for $c\in\F_8^*$). So $\phi^3$ acts trivially on each vertex set, while $\phi$ itself moves labels ($\omega\mapsto\omega^2$). \emph{The order of $\Gamma_8$.} Every generator maps a labelled vertex $(X,c)$ to $(\bar\gamma X,\hat\gamma c)$ and an unlabelled vertex $X$ to $\bar\gamma X$, where $\bar\gamma$ is the induced permutation of $\Pone$ (and so of $\Ptwo$ and $\Eset$) and $\hat\gamma$ that of the labels. On $\Pone$ the generators induce $\PSL(2,8)$ and the Frobenius map, which generate $\PGaL(2,8)$ ($t$ and $\iota$ act trivially). On the exponents $e$ of labels $c=\omega^e$, the maps $t$, $\phi$, $\iota$ act as $e\mapsto e+1$, $e\mapsto2e$, $e\mapsto-e$, and $\SL(2,8)$ acts trivially; these generate $\mathrm{AGL}(1,7)$, of order $42$. So $\gamma\mapsto(\bar\gamma,\hat\gamma)$ is a homomorphism of $\Gamma_8$ into $\PGaL(2,8)\times\mathrm{AGL}(1,7)$, and it is injective, since an element with $\bar\gamma=1$ and $\hat\gamma=1$ fixes every vertex. Let $\epsilon_1\colon\PGaL(2,8)\to\PGaL(2,8)/\PSL(2,8)\cong C_3$ and $\epsilon_2\colon\mathrm{AGL}(1,7)\to(\Z/7)^*/\{\pm1\}\cong C_3$ (the class of the linear part), identified so that the Frobenius map and multiplication by $2$ go to the same generator. Each generator lies in the fibre product $F=\{(x,y):\epsilon_1(x)=\epsilon_2(y)\}$, of order $1512\cdot42/3=21168$. Conversely the image contains $\PSL(2,8)\times1$ (from $\SL(2,8)\cong\PSL(2,8)$), $1\times\langle e\mapsto e+1,\ e\mapsto-e\rangle\cong1\times D_{14}$, and the element $(\text{Frobenius},\,e\mapsto2e)$, which maps onto a generator of $F/(\PSL(2,8)\times D_{14})\cong C_3$; so the image is $F$ and $|\Gamma_8|=504\cdot7\cdot3\cdot2=21168$. The action on each vertex set is faithful: an element acting trivially fixes every label, so $\hat\gamma=1$, and fixes every member of $\Ptwo$ (resp.\ $\Eset$), so $\bar\gamma$ lies in the kernel of the action of $\PGaL(2,8)$ on $\Ptwo$ (resp.\ $\Eset$). That kernel is trivial: on $\Ptwo$ because a point is the intersection of two pairs; on $\Eset$ because the kernel is a normal subgroup meeting the simple group $\PSL(2,8)$ trivially ($\PSL(2,8)$ moves elements of $\Eset$), hence it centralises $\PSL(2,8)$, whose centraliser in $\PGaL(2,8)$ is trivial. The same argument with $\hat\gamma$ restricted to $\langle e\mapsto2e,\,e\mapsto-e\rangle$ gives $|\langle\SL(2,8),\phi,\iota\rangle|=504\cdot3\cdot2=3024$. In this group $\iota$ is central, $\langle\SL(2,8),\phi\rangle \cong\PGaL(2,8)$, and $\iota\notin\langle\SL(2,8),\phi\rangle$ (the elements of $\langle\SL(2,8),\phi\rangle$ that fix every point have $\epsilon_1=1$, hence lie in $\SL(2,8)$, which fixes every label), so the group is $C_2\times\PGaL(2,8)$. (3) $\SL(2,8)$ is transitive on $\Ptwo$ (it is $3$-transitive on $\Pone$) and on $\PG(1,64)\setminus\PG(1,8)$ (a point stabiliser is a non-split torus $C_9$, of index $56$), hence on $\Eset$; $t$ is transitive on labels. For the product graphs the label $1$ is fixed by $\phi$ and $\iota$, and $\langle\phi,\iota\rangle$ is transitive on $\F_8^*\setminus\{1\}$. \end{proof} For $\Gamma^\star_{261}$ we can check the rows of the points by hand. \begin{proposition}\label{prop:points261} In $\Gamma^\star_{261}$, $\star\in\{Q,P\}$, two adjacent vertices of which at least one is a point have exactly $14$ common neighbours, and a point and a non-adjacent vertex have exactly $16$. The local graph of a point in $\Gamma^Q_{261}$ is the $8\times8$ rook's graph $L_2(8)$, an $\srg(64,14,6,2)$. \end{proposition} \begin{proof} Two points $p,p'$: the other $7$ points and the $7$ copies of $\{p,p'\}$. A point $p$ and $(\pi,c)$ with $p\in\pi$: the other point of $\pi$, the other $6$ copies of $\pi$, and for each of the $7$ pairs $\pi'\ni p$, $\pi'\neq\pi$, the unique copy $(\pi',c')$ with $d=1$. A point $p$ and $(\pi,c)$ with $p\notin\pi$: the $2$ points of $\pi$; the $2$ pairs $\{p,a\}$, $a\in\pi$, with one copy each; and the $6$ pairs $\{p,b\}$, $b\notin\pi$, disjoint from $\pi$, with two copies each: $2+2+12=16$. Under the quotient rule, $(\{p,a\},c)\sim(\{p,b\},c')$ for $a\ne b$ iff $c=c'$. So the $64$ neighbours of $p$ form an $8\times8$ array with rows indexed by $a\in\Pone\setminus\{p\}$ (row $a$: the point $a$ and the $7$ copies of $\{p,a\}$) and columns indexed by $\F_8^*\cup\{\text{point}\}$, and two neighbours are adjacent iff they share a row or a column. \end{proof} \begin{computation}\label{comp:pg18} The five graphs of Construction~\ref{con:pg18} with $\star$ as stated are strongly regular, with parameters $(261,64,14,16)$, $(280,63,14,14)$ and $(232,63,14,18)$. By Proposition~\ref{prop:inv} it suffices to compute the rows of $A^2$ for one vertex in each orbit of the group of Proposition~\ref{prop:inv}(3) (two or three vertices, plus one per label class for the product graphs); we have also computed the full matrices $A^2$ with exact integer arithmetic, with several independent programs (Section~\ref{sec:verify}). The product rule on $(\Eset\times\F_8^*)\cup \Ptwo$ is $63$-regular with $\lambda\in\{14,18\}$, hence not strongly regular. \end{computation} \begin{remark} Several cliques and cocliques can be read off. In both $\Gamma^\star_{261}$ the $9$ points, and the $7$ copies of a pair together with its $2$ points, are cliques of size $9=1-k/s$, i.e.\ Delsarte cliques; in $\Gamma^Q_{261}$ so is a point $p$ together with the $8$ copies $(\{p,a\},c)$ with one fixed label $c$ (a column of the grid of Proposition~\ref{prop:points261}). In $\Gamma^\star_{280}$ the set $\Eset$ is a coclique of size $28=v(-s)/(k-s)$, a Hoffman coclique. In $\Gamma_{232}$ the $8$ pairs through a point form a Delsarte $8$-clique of $T(9)$. \end{remark} \subsection{\texorpdfstring{The first graph on $160$ vertices: a grid and flags}{The first graph on 160 vertices: a grid and flags}}\label{sec:160} Let $R=\{1,2,3,4\}$ and consider the $16$ cells $(i,j)\in R\times R$ of a $4\times4$ grid. Two distinct cells are \emph{collinear} if they share a row or a column. For an even permutation $s\in A_4$ the \emph{transversal} $T_s=\{(j,s(j)):j\in R\}$ is a set of four pairwise non-collinear cells; there are $12$ even transversals. A \emph{flag} is a pair $(T,f)$ with $T$ an even transversal and $f\in T$; there are $48$ flags. \begin{construction}\label{con:160} $\Gamma^{(1)}_{160}$ has as vertices the $16$ cells and the $144$ triples $(T,f,z)$ with $(T,f)$ a flag and $z\in\Z/3$. Cells are pairwise non-adjacent. A cell $x$ is adjacent to $(T,f,z)$ iff $x\notin T$ and $x$ is not collinear with $f$. For two triples $(T,f,z)$, $(T',f',z')$ put $\delta=z'-z$. They are adjacent in exactly the following cases: \begin{enumerate} \item $|T\cap T'|=1$, $f\neq f'$, $f$ and $f'$ collinear, $f\notin T'$, $f'\notin T$, and $\delta=0$; \item $|T\cap T'|=1$, $f\neq f'$ not collinear, $f\notin T'$, $f'\notin T$, and $\delta\neq0$; \item $|T\cap T'|=1$, $f\neq f'$ not collinear, and exactly one of $f\in T'$, $f'\in T$ holds. Order the two triples so that $f\in T'$ and $f'\notin T$; then they are adjacent iff $\delta=z'-z=2$. Equivalently, for the ordering with $f\notin T'$ and $f'\in T$, they are adjacent iff $\delta=1$; \item $T\cap T'=\emptyset$, $f$ and $f'$ collinear, and $\delta\neq0$. \end{enumerate} All other pairs of triples are non-adjacent; in particular two triples with $T=T'$, or with $|T\cap T'|=1$ and $f=f'$ (the common cell), are non-adjacent. \end{construction} Two even transversals meet in $4$, $1$ or $0$ cells, since a non-trivial element of $A_4$ fixes one point or none. If $|T\cap T'|=1$ and $f\neq f'$, then $f\in T'$ and $f'\in T$ cannot both hold (the two transversals would share $f$ and $f'$), so (2) and (3) cover all non-collinear $f\ne f'$. The rule is symmetric: exchanging the two triples negates $\delta$ and exchanges the two memberships, so the two formulations of (3) describe the same unordered pairs. Exchanging the values $2$ and $1$ in (3) gives an isomorphic graph: it is the image under $z\mapsto-z$, which negates $\delta$ and preserves (1), (2) and (4). A cell lies on $3$ even transversals. So $9$ even transversals avoid a given cell $x$; each of them meets the row and the column of $x$ in two different cells, so it has exactly $2$ cells not collinear with $x$, and $x$ has $9\cdot2\cdot3=54$ neighbours. The cells form a coclique of size $16$, which is the Hoffman bound. Let $W\le S_4\wr S_2$ be the group of maps $(i,j)\mapsto(i^a,j^b)$ with $a,b\in S_4$ of the same sign, together with the transposition $\tau\colon(i,j)\mapsto(j,i)$; $|W|=2\cdot24\cdot24/2=576$. On flags, $(a,b)$ maps $(T_s,(i,s(i)))$ to $(T_{a^{-1}sb},(i^a,i^{sb}))$ (the sign of $a^{-1}sb$ is that of $s$), and $\tau$ maps it to $(T_{s^{-1}},(s(i),i))$. Let $C_3$ act by $z\mapsto z+1$ on triples and trivially on cells. \begin{proposition}\label{prop:160} $W\times C_3$, of order $1728$, acts on $\Gamma^{(1)}_{160}$ as a group of automorphisms with the two orbits ``cells'' ($16$) and ``triples'' ($144$). $W$ is isomorphic to the group $2^4{:}(S_3\times S_3)$ of TomLib, and acts primitively with rank $3$ on the cells, with cell stabiliser $S_3\times S_3$. \end{proposition} \begin{proof} The rule uses only incidence, collinearity, intersection sizes and $\delta$, which $W$ preserves ($W$ acts trivially on $z$), and $\delta$ is invariant under $z\mapsto z+1$. $W$ is transitive on the cells, on the $12$ even transversals (by $s\mapsto a^{-1}sb$) and on the four cells of a transversal, hence on flags, and $C_3$ is transitive on $z$. The cell stabiliser has the three orbits ``equal'', ``collinear'' ($6$) and ``non-collinear'' ($9$), so the rank on cells is $3$. The isomorphism with the TomLib group, the primitivity and the cell stabiliser were checked in GAP (\code{paper\_grid\_group.g}). \end{proof} \begin{computation}\label{comp:160} $\Gamma^{(1)}_{160}$ is an $\srg(160,54,18,18)$ and $\Aut\Gamma^{(1)}_{160}=W\times C_3$. The rule of Construction~\ref{con:160} was extracted from the orbital fusion found by the search (Section~\ref{sec:search}) by computing, for every ordered pair of vertices, the invariants used in the rule: adjacency is a function of these invariants, with no conflict over all $160\cdot159$ ordered pairs (\code{rules160.g}). The graph built from the rule alone has the same canonical form as the search hit. The independent audit (Section~\ref{sec:verify}) built the graph from the rule with both orientations of (3) and obtained the canonical form of the search hit in both cases. \end{computation} \subsection{\texorpdfstring{The other two graphs on $160$ vertices}{The other two graphs on 160 vertices}}\label{sec:160bc} The group $W=2^4{:}(S_3\times S_3)$ (TomLib table \code{2\^{}4:(S3xS3)}) has subgroups $C_6$ (class $25$), $D_{12}$ and $S_3\times S_3$ (class $105$), of indices $96$, $48$ and $16$. There are two classes of subgroups $D_{12}$, numbers $60$ and $66$, both of index $48$. In the model of Section~\ref{sec:160} the action on the cosets of $S_3\times S_3$ is the action on the cells. The search action that gave $\Gamma^{(1)}_{160}$ uses three copies of $W/D_{12}$ with $D_{12}$ of class $60$; the same action with class $66$ gives $\Gamma^{(1)}_{160}$ as well (Table~\ref{tab:fusions}). The subgroup $C_6$ of class $25$ lies in exactly one conjugate of a $D_{12}$ of class $60$, with index $2$, and in no conjugate of class $66$; so the action of $W$ on $W/C_6\sqcup W/D_{12}\sqcup W/(S_3\times S_3)$, with $D_{12}$ of class $60$, has a $96$-point orbit that is a double cover of the $48$-point orbit (\code{c25.log}). On these $96+48+16=160$ points $W$ has $43$ symmetrised non-diagonal orbitals, and exactly four unions of them are $\srg(160,54,18,18)$. They fall into two isomorphism classes, $\Gamma^{(2)}_{160}$ and $\Gamma^{(3)}_{160}$, with two fusions each. The automorphism group of $\Gamma^{(3)}_{160}$ is $W$ itself, with the three orbits $96$, $48$, $16$; that of $\Gamma^{(2)}_{160}$ has order $1728$, contains $W$ with index $3$, and fuses the orbits of lengths $96$ and $48$. We have not extracted a prose rule for these two graphs. They are defined as follows, so that they can be rebuilt from TomLib without our files. Number the $160$ points as in the GAP code below (GAP 4.12.1, TomLib 1.2.11): points $1$--$96$, $97$--$144$ and $145$--$160$ are the right cosets of the representatives (\code{RepresentativeTom}) of the classes $25$, $60$ and $105$, each block in the order returned by \code{RightCosets}. A graph is given by a list of representative pairs and is the union of the orbits of $W$ on unordered pairs of points that contain one of them: \begin{align*} \Gamma^{(2)}_{160}:&\ \{1,4\},\{1,6\},\{1,9\},\{1,12\},\{1,97\},\{1,103\},\{1,110\},\{1,147\},\{97,109\},\{97,150\};\\ \Gamma^{(3)}_{160}:&\ \{1,6\},\{1,9\},\{1,12\},\{1,14\},\{1,99\},\{1,103\},\{1,104\},\{1,145\},\{97,109\},\{97,150\}. \end{align*} Each list contains one pair from each of the $10$ symmetrised orbitals of the union. \begin{lstlisting}[language={},numbers=none] LoadPackage("tomlib"); tom := TableOfMarks("2^4:(S3xS3)");; W := UnderlyingGroup(tom);; cos := Concatenation(List([25, 60, 105], i -> RightCosets(W, RepresentativeTom(tom, i))));; P := Group(List(GeneratorsOfGroup(W), g -> PermList(List(cos, c -> Position(cos, c * g)))));; graph := reps -> Set(Concatenation(List(reps, r -> Orbit(P, Set(r), OnSets))));; G2 := graph([[1,4],[1,6],[1,9],[1,12],[1,97],[1,103],[1,110],[1,147],[97,109],[97,150]]);; G3 := graph([[1,6],[1,9],[1,12],[1,14],[1,99],[1,103],[1,104],[1,145],[97,109],[97,150]]);; \end{lstlisting} The script \code{paper\_160bc.g} consists of these lines, the two other fusions and a graph6 writer; its outputs have the canonical forms of \code{G160b.g6} and \code{G160c.g6} and pass the exact check (\code{canon\_160bc.log}). The graphs were also rebuilt from the orbital lists of the fusion enumeration in Section~\ref{sec:verify}, and the independent audit rebuilt the action from the TomLib table with its own orbital extractor, enumerated all fusions with its own solver model, and again found exactly four fusions in two classes, with the canonical forms of \code{G160b.g6} and \code{G160c.g6}. \section{The groups}\label{sec:groups} \subsection{\texorpdfstring{$\PGaL(2,8)$ and $\Gamma_8$}{PGammaL(2,8) and Gamma8}} TomLib contains the table of marks of $G=L_2(8){:}3\times2=C_2\times\PGaL(2,8)$, of order $3024$; in this section $G$ always denotes this group. Its central involution generates the factor $C_2$; in the model of Section~\ref{sec:pg18} it is $\iota$. The subgroups that occur are listed in Table~\ref{tab:tomlib} (\code{groups.log}). The point stabiliser $C_2\times\mathrm{A\Gamma L}(1,8)$ gives $\Pone$, the normalisers of the split and non-split tori give $\Ptwo$ and $\Eset$ (each element of $\Eset$ is the pair of fixed points in $\PG(1,64)$ of a Sylow $3$-subgroup $C_9$ of $\PSL(2,8)$, a non-split torus), and the dihedral subgroups $D_{14}$ and $D_{18}$ of $\PSL(2,8)$, which do not contain the centre, give $6$ copies of $\Ptwo$ resp.\ $\Eset$, permuted regularly by $G/\PSL(2,8)\cong C_6$. \begin{table}[ht] \caption{Subgroups of $G=C_2\times\PGaL(2,8)$ (TomLib table \code{L2(8):3x2}) used by the actions, and the coset spaces. ``$Z$'' is the centre of $G$.}\label{tab:tomlib} \centering\small \begin{tabular}{@{}rllrl@{}} \toprule class & $H$ & remark & index & $G/H$\\ \midrule $57$ & $C_2\times\mathrm{A\Gamma L}(1,8)$ & point stabiliser & $9$ & $\Pone$ ($Z$ acts trivially)\\ $53$ & $C_2\times(C_7{:}C_6)$ & normaliser of a split torus & $36$ & $\Ptwo$\\ $54$ & $C_2\times(C_9{:}C_6)$ & normaliser of a non-split torus & $28$ & $\Eset$\\ $27$ & $D_{14}\le\PSL(2,8)$ & $Z\not\le H$; in class $53$ & $216$ & $C_6\times\Ptwo$\\ $31$ & $D_{18}\le\PSL(2,8)$ & $Z\not\le H$; in class $54$ & $168$ & $C_6\times\Eset$\\ \bottomrule \end{tabular} \end{table} The three actions of $G$ found by the search are on $G/H_{31}\sqcup G/H_{53}\sqcup G/H_{54}$ ($168+36+28=232$), $G/H_{27}\sqcup G/H_{53}\sqcup G/H_{57}$ ($216+36+9=261$) and $G/H_{27}\sqcup G/H_{53}\sqcup G/H_{54}$ ($216+36+28=280$). Identify the $1+6$ copies of $\Ptwo$ (or $\Eset$) with $\Ptwo\times\F_8^*$ (or $\Eset\times\F_8^*$): the copy with label $1$ is the orbit $G/H_{53}$ (or $G/H_{54}$), and the six copies with labels $c\neq1$ are permuted by $G/\PSL(2,8)\cong C_6$ as $\phi$ and $\iota$ permute $\F_8^*\setminus\{1\}$. This gives the subgroup $\langle\SL(2,8),\phi,\iota\rangle$ of $\Gamma_8$ of Proposition~\ref{prop:inv}; the label shift $t$ is the extra symmetry that fuses the orbit of label $1$ with the six others. A fusion search on $\Gamma_8$ and on this subgroup in the explicit model (\code{gamma\_family.g}) re-finds all five classes with the same canonical forms. \subsection{Automorphism groups}\label{sec:aut} The orders of the automorphism groups were computed with \code{dreadnaut} \cite{nauty}; the generators were checked to be automorphisms in GAP, which also computed the structure (\code{aut\_struct.log}, \code{aut\_struct\_160bc.log}). \emph{The quotient graphs.} $|\Aut\Gamma_{232}|=|\Aut\Gamma^Q_{261}|=|\Aut\Gamma^Q_{280}|=21168=|\Gamma_8|$. Since $\Gamma_8\le\Aut$ (Proposition~\ref{prop:inv}(2), which also proves $|\Gamma_8|=21168$), $\Aut=\Gamma_8$ for all three. As a further check, the audit built $\Gamma_8$ from independently written generators and verified in GAP that it is isomorphic to $\Aut\Gamma^Q_{261}$. GAP gives the structure $C_7{:}(C_2\times(\PSL(2,8){:}C_3))$, derived subgroup $\GL(2,8)\cong\SL(2,8)\times C_7$ and trivial centre; the three groups are isomorphic, as they must be. Abstractly $\Aut\cong\GL(2,8)\rtimes C_6$, where $C_6$ is generated by the field automorphism and by the inversion of the scalars; in our model the scalars $C_7$ act on labels, not on points. \emph{The product graphs.} $|\Aut\Gamma^P_{261}|=|\Aut\Gamma^P_{280}|=3024$, so $\Aut=C_2\times\PGaL(2,8)$, the TomLib group itself (again because this group, of order $3024$ by Proposition~\ref{prop:inv}(2), is contained in $\Aut$). \emph{The graphs on $160$ vertices.} $\Aut\Gamma^{(1)}_{160}=W\times C_3\cong C_3\times2^4{:}(S_3\times S_3)$ with centre $C_3$, acting with rank $3$ on the cells (unfaithfully: the $C_3$ fixes every cell). $\Aut\Gamma^{(2)}_{160}$ has order $1728$ and structure $C_3{:}(2^4{:}(S_3\times S_3))$ with trivial centre; it is not isomorphic to $\Aut\Gamma^{(1)}_{160}$ (GAP's \code{IsomorphismGroups}). $\Aut\Gamma^{(3)}_{160}\cong2^4{:}(S_3\times S_3)$, of order $576$. Table~\ref{tab:suborbits} lists the vertex stabilisers. \begin{table}[ht] \caption{Vertex stabilisers in $\Aut$ and the rank of $\Aut$ on each of its vertex orbits.}\label{tab:suborbits} \centering\small \begin{tabular}{@{}ll@{}} \toprule graph & orbit: stabiliser (rank of $\Aut$ on the orbit)\\ \midrule $\Gamma^{(1)}_{160}$ & $144$: $D_{12}$ ($27$); \ $16$: $C_3\times S_3\times S_3$ ($3$)\\ $\Gamma^{(2)}_{160}$ & $144$: $D_{12}$ ($19$); \ $16$: $((C_3\times C_3){:}C_2)\times S_3$ ($3$)\\ $\Gamma^{(3)}_{160}$ & $96$: $C_6$ ($20$); \ $48$: $D_{12}$ ($9$); \ $16$: $S_3\times S_3$ ($3$)\\ $\Gamma_{232}$ & $196$: $C_2\times(C_9{:}C_6)$ ($6$); \ $36$: $(C_7\times C_7){:}(C_6\times C_2)$ ($3$)\\ $\Gamma^{P}_{261}$ & $216$: $D_{14}$ ($30$); \ $36$: $C_2\times(C_7{:}C_6)$ ($3$); \ $9$: $C_2\times\mathrm{A\Gamma L}(1,8)$ ($2$)\\ $\Gamma^{Q}_{261}$ & $252$: $C_2\times(C_7{:}C_6)$ ($8$); \ $9$: $(((C_2^3){:}C_7)\times D_{14}){:}C_3$ ($2$)\\ $\Gamma^{P}_{280}$ & $216$: $D_{14}$ ($30$); \ $36$: $C_2\times(C_7{:}C_6)$ ($3$); \ $28$: $C_2\times(C_9{:}C_6)$ ($2$)\\ $\Gamma^{Q}_{280}$ & $252$: $C_2\times(C_7{:}C_6)$ ($8$); \ $28$: $C_7{:}(C_2\times(C_9{:}C_6))$ ($2$)\\ \bottomrule \end{tabular} \end{table} \section{Structure}\label{sec:structure} Table~\ref{tab:props} collects invariants that separate the graphs and decide geometricity. All entries are computer results (\code{structure.py}, \code{pranks.g}, \code{geom.py}, \code{geom\_check2.py}, \code{set\_orbits.g}). The independent audit recomputed every entry with its own programs (\code{b4\_invariants.py}, \code{cliques.c}) and found the same values; it also supplied the complete count $0$ for the Hoffman cocliques of $\Gamma^P_{261}$ and the complete counts $37$ and $289$ for the two $280$ graphs. The referee's enumerator reproduced all coclique counts once more. \begin{table}[ht] \caption{Invariants. $\mathrm{rk}_2(M)$ is the rank of $M$ over $\F_2$. For $(160,54,18,18)$ and $(261,64,14,16)$ we have $A^2\equiv0$, for $(232,63,14,18)$ and $(280,63,14,14)$ we have $A^2\equiv I\pmod 2$, so in each row one of $A$, $A+I$ is invertible over $\F_2$ (it squares to $I$) and the other carries the information. $\mathrm{rk}_3$ resp.\ $\mathrm{rk}_7$ is the rank of $A-rI$ over $\F_3$ resp.\ $\F_7$ (equal to that of $A-sI$ when $p\mid r-s$). ``Del.'' is the number of Delsarte cliques, with its decomposition into $\Aut$-orbits; ``Hoff.'' the number of Hoffman cocliques, with the number of $\Aut$-orbits or their lengths (all counts complete). ``loc.'' lists the number of distinct eigenvalues of the local graph of a vertex in each $\Aut$-orbit, in the order of Table~\ref{tab:graphs}.}\label{tab:props} \centering\small \begin{tabular}{@{}lrrrlll@{}} \toprule graph & $\mathrm{rk}_2(A)$ & $\mathrm{rk}_2(A{+}I)$ & $\mathrm{rk}_3$/$\mathrm{rk}_7$ & Del. & Hoff. & loc.\\ \midrule $\Gamma^{(1)}_{160}$ & $54$ & $160$ & $\mathrm{rk}_3=60$ & $0$ & $158$ ($7$ orbits) & $34$, $10$\\ $\Gamma^{(2)}_{160}$ & $34$ & $160$ & $\mathrm{rk}_3=60$ & $0$ & $242$ ($10$ orbits) & $36$, $6$\\ $\Gamma^{(3)}_{160}$ & $44$ & $160$ & $\mathrm{rk}_3=60$ & $0$ & $186$ ($12$ orbits) & $36$, $36$, $18$\\ $\Gamma_{232}$ & $232$ & $64$ & $\mathrm{rk}_7=59$ & $513=504+9$ & $0$ & $6$, $5$\\ $\Gamma^{P}_{261}$ & $116$ & $261$ & $\mathrm{rk}_7=87$ & $46=36+9+1$ & $0$ & $40$, $15$, $5$\\ $\Gamma^{Q}_{261}$ & $44$ & $261$ & $\mathrm{rk}_7=87$ & $100=63+36+1$ & $252=126+126$ & $14$, $3$\\ $\Gamma^{P}_{280}$ & $280$ & $136$ & $\mathrm{rk}_7=88$ & $0$ & $37=36+1$ & $36$, $12$, $12$\\ $\Gamma^{Q}_{280}$ & $280$ & $64$ & $\mathrm{rk}_7=88$ & $0$ & $289=126{+}126{+}36{+}1$ & $14$, $8$\\ \bottomrule \end{tabular} \end{table} \subsection{Delsarte cliques and geometricity} The Delsarte cliques were enumerated completely: every clique of size $c=1-k/s$ is an edge $\{u,w\}$ together with a $(c-2)$-clique in the common neighbourhood of $u$ and $w$, and \code{geom.py} lists all of them and checks that no clique of size $c+1$ exists. Whether a subfamily partitions the edges was decided twice, by CP-SAT (\code{geom.py}) and by an exact-cover search with Knuth's Algorithm~X (\code{geom\_check2.py}). The results: \begin{itemize} \item $\Gamma^{(i)}_{160}$, $\Gamma^P_{280}$, $\Gamma^Q_{280}$ (Delsarte bound $10$) have no clique of size $10$ at all. So no edge lies in a Delsarte clique, and none of these graphs is geometric. For the $160$ graphs this also follows from the non-existence of $\mathrm{pg}(9,5,3)$. \item $\Gamma^P_{261}$ has $46$ cliques of size $9$: the $9$ points; the $36$ sets ``$7$ copies of a pair and its $2$ points''; and $9$ sets ``a point $p$ and the $8$ pairs $(\{p,a\},1)$''. $\Gamma^Q_{261}$ has $100$ cliques of size $9$: the $9$ points; the $36$ sets ``$7$ copies of a pair and its $2$ points''; and the $63$ columns ``a point $p$ and the $8$ pairs $(\{p,a\},c)$'' of Proposition~\ref{prop:points261}. A $\mathrm{pg}(8,7,2)$ on $261$ points has $261\cdot8/9=232$ lines, all of them Delsarte cliques of its collinearity graph. Since $46<232$ and $100<232$, neither graph is geometric. (Equivalently, $6804$ resp.\ $5292$ of the $8352$ edges lie in no Delsarte clique.) \item $\Gamma_{232}$ has $513$ cliques of size $8$, in two $\Aut$-orbits: the $9$ ``stars'' of $T(9)$ (the $8$ pairs through a point) and $504$ cliques consisting of $7$ vertices of $\Eset\times\F_8^*$ and one pair, with setwise stabiliser $C_7{:}C_6$. Together they cover every edge: $252$ edges once and $7056$ twice. This alone does not exclude a partition: a $\mathrm{pg}(7,8,2)$ would need $232\cdot9/8=261$ of the cliques partitioning the $7308$ edges, and such a subfamily must contain the unique clique through each of the $252$ edges covered once, but the counts do not decide whether the remaining edges can be covered exactly once. This was decided by the exact-cover searches: CP-SAT (\code{geom.py}) and Algorithm~X (\code{geom\_check2.py}), both run to completion, and independently the audit's CP-SAT model, all report \code{INFEASIBLE}, i.e.\ no subfamily of the $513$ cliques partitions the edge set. So $\Gamma_{232}$ is not geometric. \end{itemize} The same programs, run on the collinearity graphs of $\mathrm{GQ}(2,2)$, $\mathrm{GQ}(2,4)$ and on the lattice graph $L_2(5)$, find the lines and the partition (\code{geom\_controls.log}). For $(160,54,18,18)$ the conclusion was forced, since $\mathrm{pg}(9,5,3)$ does not exist \cite{BrouwerTable}. The graphs therefore do not decide the existence of $\mathrm{pg}(7,8,2)$, $\mathrm{pg}(8,7,2)$ and $\mathrm{pg}(9,6,2)$. \subsection{Hoffman cocliques} In $\Gamma^{(i)}_{160}$ the cells, and in $\Gamma^\star_{280}$ the set $\Eset$, are Hoffman cocliques by construction. The Hoffman cocliques were enumerated with CP-SAT \cite{ORTools} in all-solutions mode, with a limit of $1800$\,s per graph. The model asks for a coclique of size $h=v(-s)/(k-s)$ in which every outside vertex has exactly $kh/(v-h)$ neighbours, a condition every Hoffman coclique satisfies (\code{structure.py}): \begin{itemize} \item for $\Gamma^{(1)}_{160}$, $\Gamma^{(2)}_{160}$, $\Gamma^{(3)}_{160}$ and $\Gamma^Q_{261}$ the enumeration finished (status \code{OPTIMAL}) with $158$, $242$, $186$ and $252$ cocliques; \item for $\Gamma_{232}$ the model is \code{INFEASIBLE}: $\Gamma_{232}$ has no coclique of size $29$; \item for $\Gamma^P_{261}$ CP-SAT returned \code{UNKNOWN} after $1800$\,s, without finding a coclique of size $29$. The independent audit decided this case with its own branch-and-bound enumerator (\code{cliques.c}, which prunes with the Hoffman neighbour count): the search finished completely, after $1\,552\,209$ nodes, with no coclique of size $29$ (\code{out/\brk audit\_twoorbit8\_fable/\brk cocl\_more.log}). The same program reproduces the complete counts $158$, $242$, $186$, $252$ and $0$ above. So $\Gamma^P_{261}$ has no Hoffman coclique; \item for $\Gamma^P_{280}$ and $\Gamma^Q_{280}$ CP-SAT did not finish within $1800$\,s (status \code{FEASIBLE}, $25$ resp.\ $47$ cocliques listed). The audit's enumerator finished both cases completely ($8\,431\,987$ resp.\ $12\,635\,254$ nodes, \code{cocl\_more.log}): $\Gamma^P_{280}$ has exactly $37$ and $\Gamma^Q_{280}$ exactly $289$ Hoffman cocliques. Under $\Aut$ they form the orbits $36+1$ resp.\ $126+126+36+1$ (\code{cocl280\_orbits.g}, on the complete lists); the orbit of length $1$ is $\{\Eset\}$ in both graphs. \end{itemize} The referee's own enumerator, a third program, reproduced all these counts completely: $158$, $242$, $186$, $0$, $0$, $252$, $37$ and $289$ for $\Gamma^{(1)}_{160}$, $\Gamma^{(2)}_{160}$, $\Gamma^{(3)}_{160}$, $\Gamma_{232}$, $\Gamma^P_{261}$, $\Gamma^Q_{261}$, $\Gamma^P_{280}$, $\Gamma^Q_{280}$. In $\Gamma^Q_{261}$ each of the $252$ cocliques contains exactly one point and $28$ labelled pairs, one on each pair avoiding that point. By hand: the points form a clique, so a Hoffman coclique $S$ contains at most one point. If it contains none, each of the $9$ points has exactly $kh/(v-h)=64\cdot29/232=8$ neighbours in $S$, but each of the $29$ labelled vertices of $S$ is adjacent to only $2$ points, and $2\cdot29=58<72$. If $S$ contains the point $p$, its $28$ labelled vertices are non-adjacent to $p$, so they lie on the $28$ pairs avoiding $p$, and at most one on each pair, since the copies of a pair form a clique. \subsection{Local graphs} The local graph of a vertex $x$ (the subgraph induced on its neighbours) is $\lambda$-regular. In $\Gamma^Q_{261}$ the local graph of a point is the rook's graph $L_2(8)$ (Proposition~\ref{prop:points261}); it is the only local graph with three eigenvalues; all others have between $5$ and $40$ distinct eigenvalues and are not strongly regular. The numbers of distinct eigenvalues are in Table~\ref{tab:props}. The smallest ones are $5$ (the points of $\Gamma^P_{261}$: $14$, $6$, $0$, $-2$, $-8$; the pairs of $\Gamma_{232}$), $6$ (the cells of $\Gamma^{(2)}_{160}$: $18$, $6$, $3$, $0$, $-3$, $-6$; the vertices of $\Eset\times\F_8^*$ in $\Gamma_{232}$) and $8$ (the vertices of $\Eset$ in $\Gamma^Q_{280}$). In $\Gamma_{232}$ the local graph of a pair has the spectrum $14^1\,5^{14}\,(2.873)^6\,(-2)^{36}\,(-4.873)^6$, and that of a vertex of $\Eset\times\F_8^*$ the spectrum $14^1\,5^{16}\,0^9\,(-1)^{12}\,(-3)^{18}\,(-4)^7$. \subsection{Non-isomorphism} Within each row the graphs are pairwise non-isomorphic: their nauty canonical forms differ (Section~\ref{sec:verify}), and so do the invariants of Table~\ref{tab:props}. For $(160,54,18,18)$ the $2$-ranks $54$, $34$, $44$ already distinguish the three graphs; for $261$ and $280$ the orders of the automorphism groups do, and so do the numbers $0$ and $252$ (for $261$) and $37$ and $289$ (for $280$) of Hoffman cocliques; for $280$ also the $2$-ranks $\mathrm{rk}_2(A+I)=136$ and $64$. The graphs were found by different search runs and are not related by switching as far as we know; we have not tested Godsil--McKay or other switchings between them. \begin{remark} For $\lambda=\mu$, i.e.\ for $\srg(160,54,18,18)$ and $\srg(280,63,14,14)$, the adjacency matrix is the incidence matrix of a symmetric $2$-$(160,54,18)$ resp.\ $2$-$(280,63,14)$ design with a polarity without absolute points. We have not studied these designs. \end{remark} \section{How the graphs were found}\label{sec:search} \subsection{Tables of marks} The \emph{table of marks} of a finite group $G$, introduced by Burnside \cite{Burnside}, is the square matrix, indexed by the conjugacy classes of subgroups, whose entry $(H,K)$ is the number of fixed points of $K$ on the coset space $G/H$. It determines, without constructing any permutation action, the lengths of the suborbits of $G$ on $G/H\times G/K$: the stabiliser $H$ of a point of $G/H$ acts on $G/K$, and its orbit lengths can be read off from products of rows of the table (Ivanov, Klin, Tsaranov and Shpektorov \cite{IKTS}). GAP's TomLib \cite{TomLib} contains the tables of marks of $414$ groups, among them $C_2\times\PGaL(2,8)$ (\code{L2(8):3x2}) and $2^4{:}(S_3\times S_3)$ (\code{2\^{}4:(S3xS3)}), with the function \code{DecomposedFixedPointVector} that performs this computation. \subsection{The search} In \cite{KovalMO} we searched for strongly regular graphs that are unions of symmetrised orbitals of a group $G$ acting on $G/H_1\sqcup G/H_2$. The search behind the present paper ran on three servers ($3\times48$ cores) from 11 October 2026, in a public Docker image built for this purpose, and covered four kinds of actions, for all $414$ TomLib groups and all $1393$ parameter sets marked ``?'' in \cite{BrouwerTable} on 10 October 2026 (for the two-orbit kinds, the $1323$ with integral eigenvalues): \begin{itemize} \item \emph{P2}: two orbits $G/H_i\sqcup G/H_j$, at higher ranks than in \cite{KovalMO}; \item \emph{T3}: three orbits $G/H_i\sqcup G/H_j\sqcup G/H_k$, $i\le j\le k$, subgroups at any depth; \item \emph{C2}, \emph{C3}: $G\times C_m$, $m\in\{2,3\}$, on two orbits, each of which is $G/H$ with $C_m$ acting trivially, $m$ copies of $G/H$ permuted cyclically by $C_m$, or $G/H_0$ with $C_m$ acting through $N_G(H_0)/H_0$. \end{itemize} Each (action, parameter set) pair was decided exactly in three steps. First, the partition into $G$-orbits of a $G$-invariant graph is equitable, and its quotient matrix must have integer entries, row sums $k$, entries that are sums of suborbit lengths, and eigenvalues other than $k$ in $\{r,s\}$; for three orbits the trace and the sum of the $2\times2$ principal minors are checked. Second, an exact subset-sum test asks whether the suborbit lengths can produce the required degrees. Third, for the survivors the permutation action and all structure constants $p^c_{ab}$ of the coherent configuration are built, and the quadratic system \[ \sum_{a,b}x_ax_b\,p^c_{ab}=\lambda x_c+\mu(1-x_c)\quad\text{for every non-diagonal class }c, \] with a $0/1$ variable $x_a$ per symmetrised orbital and the degree equations, is solved by the CP-SAT solver of OR-Tools \cite{ORTools} with a limit of $30$\,s; every solution is expanded to the full matrix and checked by exact integer arithmetic. Each start of the image first runs must-find controls: the graph $\srg(576,253,112,110)$ of \cite{KovalMO} from two different actions, $\srg(1024,495,238,240)$, the Higman--Sims graph from a two-orbit action of $S_7$ and from the three-orbit action of $M_{22}$ on $1+22+77$ points, the complement of the Schl\"afli graph, and a planted non-strongly-regular graph that must be rejected. The sweep starts only if all controls pass. The search is not complete: actions are limited by a rank bound, groups without a TomLib table are not covered, and neither are four or more orbits apart from the $G\times C_m$ cases. Nothing in this paper depends on its completeness. \subsection{The hits, and all fusions of the same actions}\label{sec:hits} The servers reported five graphs on the four open rows: $\srg(232,63,14,18)$, $\srg(261,64,14,16)$ and $\srg(280,63,14,14)$ from the T3 actions of $C_2\times\PGaL(2,8)$ of Table~\ref{tab:tomlib}; $\srg(160,54,18,18)$ from the C3 action of $2^4{:}(S_3\times S_3)\times C_3$ on three copies of $W/D_{12}$ and $W/(S_3\times S_3)$ ($3\cdot48+16$); and a second $\srg(160,54,18,18)$ from the T3 action of $2^4{:}(S_3\times S_3)$ on $96+48+16$ points (Section~\ref{sec:160bc}). The decider stops at the first solution of each (action, parameter set) pair. We therefore rebuilt every hit action from scratch in GAP (orbitals as orbits on ordered pairs, without the search code) and enumerated \emph{all} unions of symmetrised orbitals with the given parameters, with CP-SAT in all-solutions mode; each solution was checked exactly and classified by canonical form (Table~\ref{tab:fusions}). This gave three graphs that the servers had not reported: $\Gamma^{(3)}_{160}$, $\Gamma^Q_{261}$ and $\Gamma^Q_{280}$. The product rule and the quotient rule of Construction~\ref{con:pg18} were then found by computing, for each fusion, the adjacency as a function of the intersection size, the cross ratio and the labels (\code{rules\_q8.g}). \begin{table}[ht] \caption{All fusions of the hit actions. ``Sym.'' is the number of symmetrised non-diagonal orbitals; all enumerations finished (status \code{OPTIMAL}, i.e.\ complete).}\label{tab:fusions} \centering\small \begin{tabular}{@{}lllrrl@{}} \toprule group & action & $v$ & sym. & fusions & classes\\ \midrule $C_2\times\PGaL(2,8)$ & T3, classes $31,53,54$ & $168+36+28$ & $28$ & $3$ & $\Gamma_{232}$\\ $C_2\times\PGaL(2,8)$ & T3, classes $27,53,57$ & $216+36+9$ & $31$ & $6$ & $\Gamma^P_{261}$ ($3$), $\Gamma^Q_{261}$ ($3$)\\ $C_2\times\PGaL(2,8)$ & T3, classes $27,53,54$ & $216+36+28$ & $33$ & $6$ & $\Gamma^P_{280}$ ($3$), $\Gamma^Q_{280}$ ($3$)\\ $2^4{:}(S_3\times S_3)\times C_3$ & C3, classes $60$ (copies), $105$ & $3\cdot48+16$ & $22$ & $2$ & $\Gamma^{(1)}_{160}$\\ $2^4{:}(S_3\times S_3)\times C_3$ & C3, classes $66$ (copies), $105$ & $3\cdot48+16$ & $22$ & $2$ & $\Gamma^{(1)}_{160}$\\ $2^4{:}(S_3\times S_3)$ & T3, classes $25,60,105$ & $96+48+16$ & $43$ & $4$ & $\Gamma^{(2)}_{160}$ ($2$), $\Gamma^{(3)}_{160}$ ($2$)\\ $\Gamma_8$ & $\Ptwo\times\F_8^*\cup\Pone$ & $252+9$ & $10$ & $3$ & $\Gamma^Q_{261}$\\ $\Gamma_8$ & $\Ptwo\times\F_8^*\cup\Eset$ & $252+28$ & $10$ & $3$ & $\Gamma^Q_{280}$\\ $\Gamma_8$ & $\Eset\times\F_8^*\cup\Ptwo$ & $196+36$ & $9$ & $3$ & $\Gamma_{232}$\\ \bottomrule \end{tabular} \end{table} \emph{The multiplicity $3$ of the $q=8$ classes.} For $m\in\{1,2,3\}$ let $\Gamma^\star_v(m)$ be the graph of Construction~\ref{con:pg18} with $\chi^{\pm1}$ replaced by $\chi^{\pm m}$ (so $\Gamma^\star_v(1)=\Gamma^\star_v$), on the same vertex set; every exponent $\pm m$ with $7\nmid m$ is one of these three up to sign. (i) $\Gamma^\star_v(m)$ is isomorphic to $\Gamma^\star_v$ via $(X,c)\mapsto(X,c^m)$ (Section~\ref{sec:pg18}). (ii) The proof of Proposition~\ref{prop:inv}(2) holds verbatim for every exponent: $\SL(2,8)$ and $t$ preserve $\chi^m$ and $d$, $\phi$ squares both, and $\iota$ inverts $d$; so $\Gamma^Q_v(m)$ is invariant under $\Gamma_8$, and $\Gamma^Q_v(m)$ and $\Gamma^P_v(m)$ are invariant under its subgroup $C_2\times\PGaL(2,8)$, i.e.\ each is a union of orbitals of the corresponding action. (iii) The three graphs $\Gamma^\star_v(1)$, $\Gamma^\star_v(2)$, $\Gamma^\star_v(3)$ are pairwise distinct as labelled graphs on the common vertex set, for each of the five pairs $(v,\star)$ (computer check by the referee with its own builder, \code{r2\_build\_q8.log}, and canonical forms in \code{r2c\_canon.log}). The $\Gamma_8$-actions of Table~\ref{tab:fusions} have exactly $3$ fusions, and the $C_2\times\PGaL(2,8)$-actions on $216+36+9$ and $216+36+28$ points exactly $3+3$ (Section~\ref{sec:groups} identifies these TomLib actions with the explicit model). By (i)--(iii) the graphs $\Gamma^\star_v(m)$, $m=1,2,3$, are three distinct fusions of each class, so they are all of them: the fusions of each $q=8$ class are exactly the graphs with exponents $\pm1$, $\pm2$, $\pm3$. For the $232$-type action of $C_2\times\PGaL(2,8)$ the product rule gives no strongly regular graph, which is why that line of the table has $3$ fusions only. The audit's own enumeration of the same actions gave the same numbers of fusions and classes. \emph{Scope of these enumerations, and why three graphs were missed.} Each line of Table~\ref{tab:fusions} is a statement about one action only: these are all unions of orbitals of that group on that vertex set with the given parameters. The servers missed $\Gamma^{(3)}_{160}$, $\Gamma^Q_{261}$ and $\Gamma^Q_{280}$ only because the decider stops at the first solution of each (action, parameter set) pair; the hit action of each of them also has a solution in another class, which was the one reported. \section{The family question}\label{sec:family} The vertex numbers of Construction~\ref{con:pg18} are polynomials in $q=8$: $\frac12(q+1)(q^2-q+2)=261$, $\frac12q(q-1)(q+2)=280$ and $\frac12q(q^2-q+2)=232$, and the parameters are those of $\mathrm{pg}(q,q-1,2)$, $\mathrm{pg}(q+1,q-2,2)$ and $\mathrm{pg}(q-1,q,2)$. Over $\PG(1,q)$, $q=2^e$, the construction makes sense with labels in $\F_q^*$ and the group $\Gamma_q=(\SL(2,q)\times C_{q-1})\rtimes(\langle\phi\rangle\times\langle\iota\rangle)$. We tested whether it continues. \begin{computation}\label{comp:family} In the explicit model over $\PG(1,q)$, with the group $\Gamma_q$, with its subgroup without the label shift $t$ (isomorphic to $C_2\times\PGaL(2,q)$), and for $q=16$ also with $\Gamma_{16}$ without $\phi$, all unions of symmetrised orbitals on the three vertex sets $(\Ptwo\times\F_q^*)\cup\Pone$, $(\Ptwo\times\F_q^*)\cup\Eset$ and $(\Eset\times\F_q^*)\cup\Ptwo$ were searched with CP-SAT: \begin{enumerate} \item $q=4$ ($v=35$, $36$, $28$): with no parameters prescribed (only $k\le(v-1)/2$ and $1\le\mu