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\title[A Counterexample to Huppert's Conjecture]{A Counterexample to Huppert's Conjecture}

\author{Shihao Wang}
\address{Shihao Wang, Qiuzhen College, Tsinghua University, Beijing, China}
\email{wangshihao25@mails.tsinghua.edu.cn}

\author{Zhiwei Zheng}
\address{Zhiwei Zheng, Yau Mathematical Sciences Center, Tsinghua University, Beijing, China}
\email{zhengzhiwei@mail.tsinghua.edu.cn}

\subjclass[2020]{Primary 20C15; Secondary 20D06}
\date{}

\begin{document}

\begin{abstract}
Let $S = \PSL_4(5)$ and let $G$ be the unique subgroup of $\PGL_4(5)$ containing $S$ with $[G:S] = 2$. We prove that $G$ and $S$ have the same set of irreducible complex character degrees. Meanwhile, $G$ is centerless and is not isomorphic to $S \times A$ for any finite abelian group $A$. This provides a counterexample to Huppert's conjecture.
\end{abstract}

\maketitle

% ================================================================
\section{Introduction}\label{sec:intro}
% ================================================================

For a finite group $H$, let $\Irr(H)$ denote the set of its irreducible complex characters and
\[
  \cd(H) = \{\chi(1) : \chi \in \Irr(H)\}
\]
the set of its irreducible character degrees, considered without multiplicities. A natural problem in representation theory is to understand how much of the structure of $H$ is encoded by the set $\cd(H)$. While $\cd(H)$ cannot distinguish between $H$ and a direct product $H \times A$ with a finite abelian group $A$, Huppert \cite[p.~829]{HuppertI} made the following conjecture:

\begin{conjecture}
Let $S$ be a nonabelian finite simple group and $H$ a finite group. If $\cd(H) = \cd(S)$, then $H \cong S \times A$ for some finite abelian group $A$.
\end{conjecture}

Huppert's conjecture has been verified for all sporadic simple groups \cite[pp.~289--290]{ADTW2013}, the alternating groups of degree at least $5$ \cite[Theorem~1.1]{BTZ2017}, and numerous families of simple groups of Lie type of low rank (see, for instance, \cite{HuppertI,NTW2012} and the references therein). For the four-dimensional projective special linear groups $\PSL_4(q)$, Nguyen, Tong-Viet, and Wakefield \cite[Theorem~1.1]{NTW2012} established the conjecture for all prime powers $q \ge 13$. For $q\equiv1\pmod4$ in this range, they show that $\cd(H)\ne\cd(\PSL_4(q))$ for every $\PSL_4(q)<H\leq\PGL_4(q)$, since the degree
\[
a(q)=\frac14(q-1)^2(q^2+q+1)(q^2+1)
\]
belongs to $\cd(\PSL_4(q))$ but not to $\cd(H)$. They explicitly note the exception at $q = 5$, where $a(5) = (5-1)(5^2+5+1)(5^2+1) = 3224$ is also a degree of $\PGL_4(5)$ \cite[proof of Proposition~6.1, case~(i)]{NTW2012}.

In this paper, we show that at $q = 5$ the unique index-two diagonal extension of $\PSL_4(5)$ inside $\PGL_4(5)$ preserves the entire set of irreducible character degrees, thereby yielding a counterexample to Huppert's conjecture.

\begin{theorem}\label{thm:counterexample}
Let $\FF_5$ denote the finite field with five elements, let $Z = \{aI_4 : a \in \FF_5^\times\}$ be the scalar subgroup of $\GL_4(5)$, and define
\[
  S = \SL_4(5)/Z = \PSL_4(5), \qquad
  G = \{A \in \GL_4(5) : \det A \in \{1, 4\}\}/Z.
\]
Then $S$ is a nonabelian finite simple group, $S \trianglelefteq G$ with $[G:S] = 2$, $Z(G) = 1$, and
\[
  \begin{aligned}
  \cd(G) = \cd(S) = \{&1, 155, 403, 496, 650, 3224, 3875, 4030, 4836, 6448,\\
  &10075, 11904, 12400, 12896, 14976, 15625, 16120, 19344, 24180\}.
  \end{aligned}
\]
In particular, $G \not\cong S \times A$ for any finite abelian group $A$.
\end{theorem}

The key is conjugation by $\overline{\delta}=\delta Z$, where $\delta=\diag(4,1,1,1)$. It moves only characters of degree $3224$, while fixed characters realize both $3224$ and $2\cdot3224=6448$. Clifford theory then gives $\cd(G)=\cd(S)$.

% ================================================================
\section{Action of the Diagonal Involution on Conjugacy Classes}\label{sec:classes}
% ================================================================

Throughout the remainder of the paper, we work over the finite field $\FF_5$ and maintain the notation of Theorem~\ref{thm:counterexample}. We write $I_4$ for the $4 \times 4$ identity matrix and $E_{ij}$ ($1 \le i, j \le 4$) for the standard matrix units. Since $a^4 = 1$ for all $a \in \FF_5^\times$, the center $Z = \{aI_4 : a \in \FF_5^\times\}$ of $\GL_4(5)$ has order $4$ and is contained in $\SL_4(5)$. For $A \in \GL_4(5)$, we denote its image in $\PGL_4(5) = \GL_4(5)/Z$ by $\overline{A} = AZ$. For a finite group $H$ and an element $h \in H$, we write $[h]_H = \{xhx^{-1} : x \in H\}$ for the conjugacy class of $h$ in $H$, and $C_H(h)$ for its centralizer in $H$.

Let $\delta=\diag(4,1,1,1)\in\GL_4(5)$. For $a \in \FF_5$, we write
\[
  J_4(a) = aI_4 + E_{12} + E_{23} + E_{34}
  = \begin{pmatrix}
      a & 1 & 0 & 0 \\
      0 & a & 1 & 0 \\
      0 & 0 & a & 1 \\
      0 & 0 & 0 & a
    \end{pmatrix}
\]
for the $4 \times 4$ Jordan block with eigenvalue $a$.

\begin{lemma}\label{lem:structure}
The group $S = \PSL_4(5)$ is nonabelian simple, $G = S \rtimes \langle \overline{\delta} \rangle$ with $[G:S] = 2$, and $C_{\PGL_4(5)}(S) = 1$. In particular, $Z(G) = 1$.
\end{lemma}

\begin{proof}
It is a standard fact that $S = \PSL_4(5)$ is a nonabelian finite simple group of order $|\SL_4(5)|/|Z| = 7\,254\,000\,000$. Since $\delta^2 = I_4$ and $\det \delta = 4 \notin (\FF_5^\times)^4 = \{1\}$, the element $\overline{\delta}$ has order $2$ in $\PGL_4(5)$, normalizes $S$, and does not belong to $S$. As $\{1, 4\}$ is the unique subgroup of order $2$ in $\FF_5^\times$, we have $G = S \rtimes \langle \overline{\delta} \rangle$ and $[G:S] = 2$.

To see that $C_{\PGL_4(5)}(S) = 1$, suppose that $\overline{A} \in \PGL_4(5)$ centralizes $S$, with representative $A \in \GL_4(5)$. For every pair of distinct indices $1 \le i \ne j \le 4$, the transvection $I_4 + E_{ij}$ lies in $\SL_4(5)$, so there exists $c_{ij} \in \FF_5^\times$ such that
\[
  A(I_4 + E_{ij})A^{-1} = c_{ij}(I_4 + E_{ij}).
\]
Comparing eigenvalues on both sides forces $c_{ij} = 1$. Hence $A$ commutes with $E_{ij}$ for all $i \ne j$, which implies that $A$ is a scalar matrix; that is, $A \in Z$ and $\overline{A} = 1$. Thus $C_{\PGL_4(5)}(S) = 1$, and since $S \le G \le \PGL_4(5)$, it follows that $Z(G) \le C_{\PGL_4(5)}(S) = 1$.
\end{proof}

We next show that conjugation by $\overline{\delta}$ fixes every conjugacy class of $S$ that does not arise from a single Jordan block of size $4$.

\begin{lemma}\label{lem:fixed-classes}
Let $A \in \SL_4(5)$. If $A$ is not $\GL_4(5)$-conjugate to $J_4(a)$ for any $a \in \FF_5^\times$, then $4 \in \det C_{\GL_4(5)}(A)$ and
\[
  [\delta A \delta^{-1}]_{\SL_4(5)} = [A]_{\SL_4(5)}, \qquad
  [\overline{\delta}\,\overline{A}\,\overline{\delta}^{-1}]_S = [\overline{A}]_S.
\]
\end{lemma}

\begin{proof}
It suffices to find $X\in C_{\GL_4(5)}(A)$ with $\det X=4$:
then $h=\delta X^{-1}\in\SL_4(5)$ satisfies
$hAh^{-1}=\delta A\delta^{-1}$.

Regard $V=\FF_5^4$ as an $\FF_5[t]$-module with $t$ acting as $A$,
and write
\[
V\cong\bigoplus_\nu \FF_5[t]/(f_\nu^{j_\nu}),
\]
where the $f_\nu$ are monic irreducible polynomials.
We will construct $X$ on one summand and let it act as the identity
on all the others.

Fix a summand $W=\FF_5[t]/(f^j)$, with $\deg f=r$.
Since $A|_W$ acts by multiplication by $t$, multiplication by any
polynomial $b(t)$ commutes with $A|_W$.
We therefore seek such a multiplication map with determinant $4$.

To determine the possible determinants, put
$K=\FF_5[t]/(f)\cong\FF_{5^r}$ and let $\beta$ be the image of $b(t)$
in $K$. If $\beta\ne0$, multiplication by $b(t)$ defines an
invertible map $T$ on $W$. The filtration
\[
W\supset fW\supset\cdots\supset f^jW=0
\]
has $j$ successive quotients isomorphic to $K$, on each of which
$T$ acts by multiplication by $\beta$. Thus
\[
\det T=N_{K/\FF_5}(\beta)^j.
\]
Consequently, to obtain $\det T=4$, it suffices to choose
$c\in\FF_5^\times$ with $c^j=4$ and then choose $\beta\in K^\times$
with $N_{K/\FF_5}(\beta)=c$, using the surjectivity of the norm map.
For $j=1,2,3$, we may take $c=4,2,4$, respectively.

Finally, the hypothesis guarantees a summand with $j\le3$:
otherwise, since $\dim V=4$, the decomposition would consist
of a single summand $\FF_5[t]/((t-a)^4)$, making $A$
conjugate to $J_4(a)$.
On such a summand choose $T$ as above and extend it by the
identity on the remaining summands. The resulting map $X$
commutes with $A$ and has determinant $4$, as required.
\end{proof}

We now analyze the remaining conjugacy classes, which come from the Jordan blocks $J_4(a)$ with $a \in \FF_5^\times$.

\begin{proposition}\label{prop:classes}
Let $N = E_{12} + E_{23} + E_{34}$, $u = I_4 + N = J_4(1)$, and $d = \diag(2,1,1,1) \in \GL_4(5)$. For $k \in \ZZ/4\ZZ$, define $u_k = d^k u d^{-k} \in \SL_4(5)$ and $\mathcal{C}_k = [\overline{u_k}]_S$. Then the following hold:
\begin{enumerate}
\item[(i)] Every conjugacy class of $S$ not fixed by conjugation by $\overline{\delta}$ belongs to $\{\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3\}$.
\item[(ii)] The four conjugacy classes $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ are pairwise distinct in $S$, and $|C_S(\overline{u_k})| = 125$ for each $k \in \ZZ/4\ZZ$.
\item[(iii)] For each $k \in \ZZ/4\ZZ$,
\[
  \overline{\delta}\mathcal{C}_k\overline{\delta}^{-1} = \mathcal{C}_{k+2} = \{s^2 : s \in \mathcal{C}_k\}.
\]
\end{enumerate}
\end{proposition}

\begin{proof}
(i) and (ii). For any $a \in \FF_5^\times$, setting $D_a = \diag(a^3, a^2, a, 1) \in \GL_4(5)$ gives $D_a J_4(a) D_a^{-1} = a J_4(1) = au$. Since $aI_4 \in Z$, we have $\overline{au} = \overline{u}$ in $S$, and hence for every $P \in \GL_4(5)$,
\[
  \overline{P J_4(a) P^{-1}} = \overline{(P D_a^{-1}) u (P D_a^{-1})^{-1}}.
\]
By Lemma~\ref{lem:fixed-classes}, any $S$-conjugacy class not fixed by conjugation by $\overline{\delta}$ has a representative of the form $\overline{P u P^{-1}}$ with $P \in \GL_4(5)$.

Since $u$ is a single Jordan block,
\[
 C_{\GL_4(5)}(u)=\{a_0I_4+a_1N+a_2N^2+a_3N^3:a_0\in\FF_5^\times,\ a_1,a_2,a_3\in\FF_5\}.
\]
This centralizer has order $4\cdot5^3=500$ and lies in $\SL_4(5)$, since its elements have determinant $a_0^4=1$.

For $P,Q\in\GL_4(5)$,
\begin{equation}\label{eq:det-conjugacy}
[PuP^{-1}]_{\SL_4(5)}=[QuQ^{-1}]_{\SL_4(5)}
\quad\Longleftrightarrow\quad \det P=\det Q.
\end{equation}
Indeed, a conjugating $h\in\SL_4(5)$ gives $Q^{-1}hP\in C_{\GL_4(5)}(u)$; conversely, take $h=QP^{-1}$. Since $\det(d^k)=2^k$, the $u_k$ represent exactly four $\SL_4(5)$-classes.

Comparing eigenvalues for both sides of $hu_ih^{-1}=cu_j$ forces $c=1$. Thus the classes $\mathcal C_k$ remain pairwise distinct in $S$, and
$C_S(\overline{u_k})=C_{\mathrm{SL}_4(5)}(u_k)/Z$ has order $500/4=125$. This completes (i) and (ii).

(iii) Since $d^2 = \diag(4,1,1,1) = \delta$, we have $\delta u_k \delta^{-1} = d^{k+2} u d^{-(k+2)} = u_{k+2}$, and therefore $\overline{\delta}\mathcal{C}_k\overline{\delta}^{-1} = \mathcal{C}_{k+2}$. To compare this with the squaring map, define
\[
  B = \begin{pmatrix}
        3 & 4 & 0 & 0 \\
        0 & 4 & 1 & 0 \\
        0 & 0 & 2 & 0 \\
        0 & 0 & 0 & 1
      \end{pmatrix} \in \GL_4(5).
\]
A direct matrix computation over $\FF_5$ gives $BN = (2N + N^2)B$, so
\[
  BuB^{-1} = I_4 + 2N + N^2 = (I_4 + N)^2 = u^2.
\]
Hence $u_k^2 = d^k u^2 d^{-k} = (d^k B) u (d^k B)^{-1}$. Since $\det B = 24 \equiv 4 = 2^2 \pmod 5$, the conjugating matrix $d^k B$ has determinant $\det(d^k B) = 2^{k+2} = \det(d^{k+2})$. By the determinant criterion~\eqref{eq:det-conjugacy}, $u_k^2$ is $\SL_4(5)$-conjugate to $u_{k+2}$. Passing to the quotient $S$, we conclude that $\{s^2 : s \in \mathcal{C}_k\} = \mathcal{C}_{k+2} = \overline{\delta}\mathcal{C}_k\overline{\delta}^{-1}$.
\end{proof}

% ================================================================
\section{Character Degrees and Proof of Theorem~\ref{thm:counterexample}}\label{sec:proof}
% ================================================================

We now determine how conjugation by $\overline{\delta}$ permutes $\Irr(S)$ using the ordinary character table of $S = \PSL_4(5)$, which is listed as \texttt{L4(5)} in the GAP Character Table Library, version~1.3.11 \cite{CTblLibL45}, and given in Appendix~\ref{app:character-table}. Throughout this section, all row labels $\chi_1, \dots, \chi_{49}$ and column indices $1, \dots, 49$ refer to the tables in Appendix~\ref{app:character-table}. For $\chi\in\Irr(S)$ and $s\in S$, write $\chi^{\overline{\delta}}(s)=\chi(\overline{\delta}^{-1}s\overline{\delta})$.

\begin{proposition}\label{prop:orbits}
Conjugation by $\overline{\delta}$ interchanges $\chi_9$ with $\chi_{10}$ and $\chi_{11}$ with $\chi_{12}$, where
\[
  \chi_9(1) = \chi_{10}(1) = \chi_{11}(1) = \chi_{12}(1) = 3224,
\]
and fixes all other $45$ irreducible characters of $S$.
\end{proposition}

\begin{proof}
The centralizer orders $c_j=|C_S(g_j)|$ listed in Tables~\ref{tab:character-table-a1}--\ref{tab:character-table-a3} in Appendix~\ref{app:character-table} show that columns $11, 12, 13, 14$ (labeled $5\mathrm{e}, 5\mathrm{f}, 5\mathrm{g}, 5\mathrm{h}$) are the unique conjugacy classes of $S$ whose centralizers have order $125$. By Proposition~\ref{prop:classes}(ii), these four columns correspond to the classes $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$. Moreover, the row $p_2(j)$ in Table~\ref{tab:character-table-a1} shows that squaring interchanges the classes indexed by $11$ and $12$, and those indexed by $13$ and $14$. By Proposition~\ref{prop:classes}(i) and (iii), conjugation by $\overline{\delta}$ induces the same permutation $(11\ 12)(13\ 14)$ on the columns of the character table and fixes all remaining $45$ columns.

In Table~\ref{tab:character-table-a1}, the $4 \times 4$ submatrix formed by rows $\chi_9, \chi_{10}, \chi_{11}, \chi_{12}$ and columns $11, 12, 13, 14$ is
\[
 \begin{gathered}
  \begin{array}{c|rrrr}
    \toprule
    & 11\ (5\mathrm{e}) & 12\ (5\mathrm{f}) & 13\ (5\mathrm{g}) & 14\ (5\mathrm{h}) \\
    \midrule
    \chi_9    & -1 & -1 & A  & A' \\
    \chi_{10} & -1 & -1 & A' & A  \\
    \chi_{11} & A  & A' & -1 & -1 \\
    \chi_{12} & A' & A  & -1 & -1 \\
    \bottomrule
  \end{array}
  \\[4pt]
  A = \frac{3 - 5\sqrt{5}}{2}, \quad A' = \frac{3 + 5\sqrt{5}}{2}.
 \end{gathered}
\]
Furthermore, $\chi_9$ and $\chi_{10}$ agree on all columns outside $\{13, 14\}$, while $\chi_{11}$ and $\chi_{12}$ agree on all columns outside $\{11, 12\}$. For every other row $\chi_i$ ($i \notin \{9, 10, 11, 12\}$), the entries in columns $11$ and $12$ are equal, and the entries in columns $13$ and $14$ are equal. Since $A \ne A'$, conjugation by $\overline{\delta}$ interchanges $\chi_9$ and $\chi_{10}$, and $\chi_{11}$ and $\chi_{12}$, while fixing all other irreducible characters of $S$. Reading column $1$ of Table~\ref{tab:character-table-a1} gives $\chi_9(1) = \chi_{10}(1) = \chi_{11}(1) = \chi_{12}(1) = 3224$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:counterexample}]
By Lemma~\ref{lem:structure}, $S = \PSL_4(5)$ is a nonabelian simple group, $G = S \rtimes \langle \overline{\delta} \rangle$ contains $S$ as a normal subgroup of index $2$, and $Z(G) = 1$. 

Reading the distinct entries in column $1$ of Table~\ref{tab:character-table-a1} yields the $19$-element set $\cd(S)$ in Theorem~\ref{thm:counterexample}.

Since $G/S$ is cyclic of prime order $2$, Clifford theory (see, e.g., \cite[Theorem~6.11 and Corollaries~6.17 and~11.22]{Isaacs}) partitions $\Irr(G)$ into two types of characters:
\begin{enumerate}
\item[(a)] Each $G$-invariant character $\chi \in \Irr(S)$ extends to two distinct irreducible characters of $G$, each of degree $\chi(1)$.
\item[(b)] The two characters in each orbit $\{\chi, \chi^{\overline{\delta}}\} \subset \Irr(S)$ of size $2$ induce to the same irreducible character $\Ind_S^G(\chi) \in \Irr(G)$ of degree $2\chi(1)$.
\end{enumerate}
By Proposition~\ref{prop:orbits}, the only characters of $S$ not fixed by conjugation by $\overline{\delta}$ are $\chi_9, \chi_{10}, \chi_{11}, \chi_{12}$, all of degree $3224$. Since $\chi_7, \chi_8 \in \Irr(S)$ also have degree $3224$ and are $G$-invariant, every degree in $\cd(S)$ is realized by at least one $G$-invariant irreducible character of $S$, so $\cd(S) \subseteq \cd(G)$. On the other hand, $\overline{\delta}$ interchanges $\chi_9$ with $\chi_{10}$ and $\chi_{11}$ with $\chi_{12}$, so these characters induce to irreducible
characters of $G$ of degree $2\cdot3224=6448$, which already belongs to $\cd(S)$ because $\chi_{16}(1) = \chi_{17}(1) = 6448$. Therefore, $\cd(G) = \cd(S) \cup \{2 \cdot 3224\} = \cd(S)$.

Finally, if $G \cong S \times A$ for some finite abelian group $A$, comparing orders gives $|A| = [G:S] = 2$, which implies $|Z(G)| = |Z(S \times A)| = |A| = 2$, a contradiction to $Z(G) = 1$.
\end{proof}

\medskip
\noindent\textbf{Acknowledgements.}
This counterexample was discovered through interactions with large language model (LLM) AI systems. The authors verified all mathematical arguments and computations and finalized the manuscript. We take full responsibility for the content of this paper.
\clearpage
\appendix
\section{The ordinary character table of \texorpdfstring{$\PSL_4(5)$}{PSL4(5)}}\label{app:character-table}
% Generated by research/build_l45_character_table_appendix.py.
% Source: the exact CTblLib export; do not edit individual entries.
\begingroup\small
The following reproduces the ordinary character table \texttt{L4(5)}
from CTblLib 1.3.11 \cite{CTblLibL45}, with its row and column order unchanged.
For a class representative $g_j$, the headers give its class name, order
$o_j$, centralizer order $c_j=|C_S(g_j)|$, and square-class number
$p_2(j)$. Column $1$, giving character degrees, is repeated in every block;
$c_1=|S|=7\,254\,000\,000$. A dot denotes zero. Noninteger values are
defined after the last block.
\par\endgroup\medskip
\renewcommand{\thetable}{A.\arabic{table}}
\setcounter{table}{0}
\refstepcounter{table}\label{tab:character-table-a1}
\noindent\textbf{Table~\thetable.} Character table of $S$:
columns $1$--$17$.\par\smallskip
\begingroup
\fontsize{8}{9.2}\selectfont
\setlength{\tabcolsep}{2.7pt}
\renewcommand{\arraystretch}{1.02}
\centering
\begin{tabular}{@{}rrrrrrrrrrrrrrrrrr@{}}
\toprule
$j$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ & $15$ & $16$ & $17$ \\
$\mathrm{class}$ & $1\mathrm{a}$ & $2\mathrm{a}$ & $3\mathrm{a}$ & $3\mathrm{b}$ & $4\mathrm{a}$ & $4\mathrm{b}$ & $5\mathrm{a}$ & $5\mathrm{b}$ & $5\mathrm{c}$ & $5\mathrm{d}$ & $5\mathrm{e}$ & $5\mathrm{f}$ & $5\mathrm{g}$ & $5\mathrm{h}$ & $6\mathrm{a}$ & $6\mathrm{b}$ & $8\mathrm{a}$ \\
$o_j$ & $1$ & $2$ & $3$ & $3$ & $4$ & $4$ & $5$ & $5$ & $5$ & $5$ & $5$ & $5$ & $5$ & $5$ & $6$ & $6$ & $8$ \\
$c_j$ & $|S|$ & $28800$ & $23400$ & $720$ & $480$ & $144$ & $375000$ & $37500$ & $37500$ & $625$ & $125$ & $125$ & $125$ & $125$ & $720$ & $72$ & $24$ \\
$p_2(j)$ & $1$ & $1$ & $3$ & $4$ & $2$ & $2$ & $7$ & $8$ & $9$ & $10$ & $12$ & $11$ & $14$ & $13$ & $4$ & $3$ & $5$ \\
\midrule
$\chi_{1}$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\
$\chi_{2}$ & $155$ & $11$ & $-1$ & $5$ & $7$ & $-1$ & $30$ & $5$ & $5$ & $5$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $5$ & $-1$ & $1$ \\
$\chi_{3}$ & $403$ & $19$ & $13$ & $1$ & $-5$ & $-1$ & $28$ & $28$ & $3$ & $3$ & $3$ & $3$ & $-2$ & $-2$ & $1$ & $1$ & $-1$ \\
$\chi_{4}$ & $403$ & $19$ & $13$ & $1$ & $-5$ & $-1$ & $28$ & $3$ & $28$ & $3$ & $-2$ & $-2$ & $3$ & $3$ & $1$ & $1$ & $-1$ \\
$\chi_{5}$ & $496$ & $16$ & $25$ & $4$ & $\cdot$ & $4$ & $-4$ & $21$ & $21$ & $-4$ & $1$ & $1$ & $1$ & $1$ & $4$ & $1$ & $\cdot$ \\
$\chi_{6}$ & $650$ & $26$ & $26$ & $-4$ & $6$ & $2$ & $25$ & $25$ & $25$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-4$ & $2$ & $\cdot$ \\
$\chi_{7}$ & $3224$ & $8$ & $26$ & $5$ & $4$ & $-4$ & $99$ & $-26$ & $-26$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $5$ & $2$ & $-2$ \\
$\chi_{8}$ & $3224$ & $8$ & $26$ & $5$ & $4$ & $4$ & $99$ & $-26$ & $-26$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $5$ & $2$ & $2$ \\
$\chi_{9}$ & $3224$ & $-8$ & $26$ & $-4$ & $\cdot$ & $\cdot$ & $-26$ & $-26$ & $49$ & $-1$ & $-1$ & $-1$ & $A$ & $A'$ & $4$ & $-2$ & $\cdot$ \\
$\chi_{10}$ & $3224$ & $-8$ & $26$ & $-4$ & $\cdot$ & $\cdot$ & $-26$ & $-26$ & $49$ & $-1$ & $-1$ & $-1$ & $A'$ & $A$ & $4$ & $-2$ & $\cdot$ \\
$\chi_{11}$ & $3224$ & $-8$ & $26$ & $-4$ & $\cdot$ & $\cdot$ & $-26$ & $49$ & $-26$ & $-1$ & $A$ & $A'$ & $-1$ & $-1$ & $4$ & $-2$ & $\cdot$ \\
$\chi_{12}$ & $3224$ & $-8$ & $26$ & $-4$ & $\cdot$ & $\cdot$ & $-26$ & $49$ & $-26$ & $-1$ & $A'$ & $A$ & $-1$ & $-1$ & $4$ & $-2$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{13}$ & $3875$ & $35$ & $-25$ & $-1$ & $11$ & $-1$ & $125$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $-1$ \\
$\chi_{14}$ & $4030$ & $46$ & $-26$ & $4$ & $-6$ & $2$ & $155$ & $5$ & $5$ & $5$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $4$ & $-2$ & $\cdot$ \\
$\chi_{15}$ & $4836$ & $-60$ & $\cdot$ & $6$ & $4$ & $\cdot$ & $211$ & $36$ & $36$ & $11$ & $1$ & $1$ & $1$ & $1$ & $6$ & $\cdot$ & $\cdot$ \\
$\chi_{16}$ & $6448$ & $16$ & $13$ & $4$ & $\cdot$ & $-4$ & $-52$ & $73$ & $-27$ & $-2$ & $-2$ & $-2$ & $3$ & $3$ & $4$ & $1$ & $\cdot$ \\
$\chi_{17}$ & $6448$ & $16$ & $13$ & $4$ & $\cdot$ & $-4$ & $-52$ & $-27$ & $73$ & $-2$ & $3$ & $3$ & $-2$ & $-2$ & $4$ & $1$ & $\cdot$ \\
$\chi_{18}$ & $10075$ & $43$ & $13$ & $-5$ & $-1$ & $-1$ & $75$ & $-50$ & $75$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-5$ & $1$ & $1$ \\
$\chi_{19}$ & $10075$ & $43$ & $13$ & $-5$ & $-1$ & $-1$ & $75$ & $75$ & $-50$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-5$ & $1$ & $1$ \\
$\chi_{20}$ & $11904$ & $\cdot$ & $48$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{21}$ & $11904$ & $\cdot$ & $48$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{22}$ & $11904$ & $\cdot$ & $48$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{23}$ & $11904$ & $\cdot$ & $-24$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{24}$ & $11904$ & $\cdot$ & $-24$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{25}$ & $11904$ & $\cdot$ & $-24$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{26}$ & $11904$ & $\cdot$ & $-24$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{27}$ & $11904$ & $\cdot$ & $-24$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{28}$ & $11904$ & $\cdot$ & $-24$ & $\cdot$ & $\cdot$ & $\cdot$ & $-96$ & $4$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{29}$ & $12400$ & $16$ & $1$ & $4$ & $\cdot$ & $4$ & $-100$ & $25$ & $25$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $4$ & $1$ & $\cdot$ \\
$\chi_{30}$ & $12896$ & $-32$ & $26$ & $8$ & $\cdot$ & $\cdot$ & $-104$ & $46$ & $46$ & $-4$ & $1$ & $1$ & $1$ & $1$ & $-8$ & $-2$ & $\cdot$ \\
$\chi_{31}$ & $12896$ & $-32$ & $-52$ & $-4$ & $\cdot$ & $\cdot$ & $-104$ & $46$ & $46$ & $-4$ & $1$ & $1$ & $1$ & $1$ & $4$ & $4$ & $\cdot$ \\
$\chi_{32}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{33}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{34}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{35}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{36}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{37}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{38}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{39}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{40}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{41}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-24$ & $-24$ & $-24$ & $1$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{42}$ & $15625$ & $25$ & $25$ & $-5$ & $5$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-5$ & $1$ & $-1$ \\
$\chi_{43}$ & $16120$ & $40$ & $-26$ & $1$ & $4$ & $4$ & $-5$ & $20$ & $20$ & $-5$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $-2$ & $-2$ \\
$\chi_{44}$ & $16120$ & $40$ & $-26$ & $1$ & $4$ & $-4$ & $-5$ & $20$ & $20$ & $-5$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $-2$ & $2$ \\
$\chi_{45}$ & $19344$ & $48$ & $\cdot$ & $6$ & $-8$ & $\cdot$ & $94$ & $-6$ & $-6$ & $-6$ & $-1$ & $-1$ & $-1$ & $-1$ & $6$ & $\cdot$ & $\cdot$ \\
$\chi_{46}$ & $19344$ & $-48$ & $\cdot$ & $-12$ & $\cdot$ & $\cdot$ & $94$ & $-6$ & $-6$ & $-6$ & $-1$ & $-1$ & $-1$ & $-1$ & $12$ & $\cdot$ & $\cdot$ \\
$\chi_{47}$ & $19344$ & $-48$ & $\cdot$ & $6$ & $\cdot$ & $\cdot$ & $94$ & $-6$ & $-6$ & $-6$ & $-1$ & $-1$ & $-1$ & $-1$ & $-6$ & $\cdot$ & $\cdot$ \\
$\chi_{48}$ & $19344$ & $-48$ & $\cdot$ & $6$ & $\cdot$ & $\cdot$ & $94$ & $-6$ & $-6$ & $-6$ & $-1$ & $-1$ & $-1$ & $-1$ & $-6$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{49}$ & $24180$ & $-12$ & $\cdot$ & $-6$ & $-4$ & $\cdot$ & $305$ & $30$ & $30$ & $5$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-6$ & $\cdot$ & $\cdot$ \\
\bottomrule
\end{tabular}\par
\endgroup
\clearpage
\refstepcounter{table}\label{tab:character-table-a2}
\noindent\textbf{Table~\thetable.} Character table of $S$: columns
$18$--$33$ (with column $1$ repeated).\par\smallskip
\begingroup
\fontsize{8}{9.2}\selectfont
\setlength{\tabcolsep}{2.7pt}
\renewcommand{\arraystretch}{1.02}
\centering
\begin{tabular}{@{}rrrrrrrrrrrrrrrrrr@{}}
\toprule
$j$ & $1$ & $18$ & $19$ & $20$ & $21$ & $22$ & $23$ & $24$ & $25$ & $26$ & $27$ & $28$ & $29$ & $30$ & $31$ & $32$ & $33$ \\
$\mathrm{class}$ & $1\mathrm{a}$ & $10\mathrm{a}$ & $10\mathrm{b}$ & $10\mathrm{c}$ & $12\mathrm{a}$ & $12\mathrm{b}$ & $12\mathrm{c}$ & $13\mathrm{a}$ & $13\mathrm{b}$ & $13\mathrm{c}$ & $15\mathrm{a}$ & $15\mathrm{b}$ & $15\mathrm{c}$ & $20\mathrm{a}$ & $24\mathrm{a}$ & $24\mathrm{b}$ & $30\mathrm{a}$ \\
$o_j$ & $1$ & $10$ & $10$ & $10$ & $12$ & $12$ & $12$ & $13$ & $13$ & $13$ & $15$ & $15$ & $15$ & $20$ & $24$ & $24$ & $30$ \\
$c_j$ & $|S|$ & $600$ & $100$ & $100$ & $36$ & $36$ & $24$ & $39$ & $39$ & $39$ & $75$ & $75$ & $30$ & $20$ & $24$ & $24$ & $30$ \\
$p_2(j)$ & $1$ & $7$ & $8$ & $9$ & $15$ & $16$ & $15$ & $25$ & $26$ & $24$ & $27$ & $28$ & $29$ & $18$ & $23$ & $23$ & $29$ \\
\midrule
$\chi_{1}$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\
$\chi_{2}$ & $155$ & $6$ & $1$ & $1$ & $-1$ & $-1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $2$ & $1$ & $1$ & $\cdot$ \\
$\chi_{3}$ & $403$ & $4$ & $4$ & $-1$ & $-1$ & $-1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-2$ & $3$ & $1$ & $\cdot$ & $-1$ & $-1$ & $1$ \\
$\chi_{4}$ & $403$ & $4$ & $-1$ & $4$ & $-1$ & $-1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $3$ & $-2$ & $1$ & $\cdot$ & $-1$ & $-1$ & $1$ \\
$\chi_{5}$ & $496$ & $-4$ & $1$ & $1$ & $-2$ & $1$ & $\cdot$ & $2$ & $2$ & $2$ & $\cdot$ & $\cdot$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{6}$ & $650$ & $1$ & $1$ & $1$ & $2$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $1$ \\
$\chi_{7}$ & $3224$ & $3$ & $-2$ & $-2$ & $-1$ & $2$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $\cdot$ & $-1$ & $1$ & $1$ & $\cdot$ \\
$\chi_{8}$ & $3224$ & $3$ & $-2$ & $-2$ & $1$ & $-2$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $\cdot$ & $-1$ & $-1$ & $-1$ & $\cdot$ \\
$\chi_{9}$ & $3224$ & $2$ & $2$ & $-3$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{10}$ & $3224$ & $2$ & $2$ & $-3$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{11}$ & $3224$ & $2$ & $-3$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{12}$ & $3224$ & $2$ & $-3$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
\addlinespace[1.5pt]
$\chi_{13}$ & $3875$ & $5$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $-1$ & $1$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $-1$ & $1$ & $-1$ & $-1$ & $-1$ \\
$\chi_{14}$ & $4030$ & $11$ & $1$ & $1$ & $2$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{15}$ & $4836$ & $-5$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $-1$ & $\cdot$ & $\cdot$ & $1$ \\
$\chi_{16}$ & $6448$ & $-4$ & $1$ & $1$ & $2$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-2$ & $3$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{17}$ & $6448$ & $-4$ & $1$ & $1$ & $2$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $3$ & $-2$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{18}$ & $10075$ & $3$ & $-2$ & $3$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-2$ & $3$ & $\cdot$ & $-1$ & $1$ & $1$ & $\cdot$ \\
$\chi_{19}$ & $10075$ & $3$ & $3$ & $-2$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $3$ & $-2$ & $\cdot$ & $-1$ & $1$ & $1$ & $\cdot$ \\
$\chi_{20}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $B$ & $D$ & $C$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{21}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $C$ & $B$ & $D$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{22}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $D$ & $C$ & $B$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{23}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $D$ & $C$ & $B$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{24}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $D$ & $C$ & $B$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{25}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $C$ & $B$ & $D$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{26}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $C$ & $B$ & $D$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{27}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $B$ & $D$ & $C$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{28}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $B$ & $D$ & $C$ & $1$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{29}$ & $12400$ & $-4$ & $1$ & $1$ & $-2$ & $1$ & $\cdot$ & $-2$ & $-2$ & $-2$ & $1$ & $1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{30}$ & $12896$ & $8$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $2$ \\
$\chi_{31}$ & $12896$ & $8$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-2$ & $-2$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ \\
$\chi_{32}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{33}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{34}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{35}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{36}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{37}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{38}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{39}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{40}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{41}$ & $14976$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{42}$ & $15625$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $\cdot$ \\
$\chi_{43}$ & $16120$ & $-5$ & $\cdot$ & $\cdot$ & $1$ & $-2$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $1$ & $-1$ & $1$ & $1$ & $1$ \\
$\chi_{44}$ & $16120$ & $-5$ & $\cdot$ & $\cdot$ & $-1$ & $2$ & $1$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $1$ & $-1$ & $-1$ & $-1$ & $1$ \\
$\chi_{45}$ & $19344$ & $-2$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $2$ & $\cdot$ & $\cdot$ & $1$ \\
$\chi_{46}$ & $19344$ & $2$ & $2$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-2$ & $\cdot$ & $\cdot$ & $\cdot$ & $2$ \\
$\chi_{47}$ & $19344$ & $2$ & $2$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $\cdot$ & $E$ & $-E$ & $-1$ \\
$\chi_{48}$ & $19344$ & $2$ & $2$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $\cdot$ & $-E$ & $E$ & $-1$ \\
\addlinespace[1.5pt]
$\chi_{49}$ & $24180$ & $-7$ & $-2$ & $-2$ & $\cdot$ & $\cdot$ & $2$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $1$ & $\cdot$ & $\cdot$ & $-1$ \\
\bottomrule
\end{tabular}\par
\endgroup
\clearpage
\refstepcounter{table}\label{tab:character-table-a3}
\noindent\textbf{Table~\thetable.} Character table of $S$: columns
$34$--$49$ (with column $1$ repeated).\par\smallskip
\begingroup
\fontsize{8}{9.2}\selectfont
\setlength{\tabcolsep}{2.7pt}
\renewcommand{\arraystretch}{1.02}
\centering
\begin{tabular}{@{}rrrrrrrrrrrrrrrrrr@{}}
\toprule
$j$ & $1$ & $34$ & $35$ & $36$ & $37$ & $38$ & $39$ & $40$ & $41$ & $42$ & $43$ & $44$ & $45$ & $46$ & $47$ & $48$ & $49$ \\
$\mathrm{class}$ & $1\mathrm{a}$ & $31\mathrm{a}$ & $31\mathrm{b}$ & $31\mathrm{c}$ & $31\mathrm{d}$ & $31\mathrm{e}$ & $31\mathrm{f}$ & $31\mathrm{g}$ & $31\mathrm{h}$ & $31\mathrm{i}$ & $31\mathrm{j}$ & $39\mathrm{a}$ & $39\mathrm{b}$ & $39\mathrm{c}$ & $39\mathrm{d}$ & $39\mathrm{e}$ & $39\mathrm{f}$ \\
$o_j$ & $1$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $39$ & $39$ & $39$ & $39$ & $39$ & $39$ \\
$c_j$ & $|S|$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $31$ & $39$ & $39$ & $39$ & $39$ & $39$ & $39$ \\
$p_2(j)$ & $1$ & $42$ & $43$ & $35$ & $34$ & $37$ & $36$ & $39$ & $38$ & $41$ & $40$ & $48$ & $49$ & $45$ & $44$ & $47$ & $46$ \\
\midrule
$\chi_{1}$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\
$\chi_{2}$ & $155$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
$\chi_{3}$ & $403$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{4}$ & $403$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{5}$ & $496$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
$\chi_{6}$ & $650$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{7}$ & $3224$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{8}$ & $3224$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{9}$ & $3224$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{10}$ & $3224$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{11}$ & $3224$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{12}$ & $3224$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{13}$ & $3875$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\
$\chi_{14}$ & $4030$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{15}$ & $4836$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{16}$ & $6448$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{17}$ & $6448$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{18}$ & $10075$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{19}$ & $10075$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{20}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $C$ & $C$ & $D$ & $D$ & $B$ & $B$ \\
$\chi_{21}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $D$ & $D$ & $B$ & $B$ & $C$ & $C$ \\
$\chi_{22}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $B$ & $B$ & $C$ & $C$ & $D$ & $D$ \\
$\chi_{23}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $K$ & $\overline{K}$ & $\overline{M}$ & $M$ & $\overline{L}$ & $L$ \\
$\chi_{24}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\overline{K}$ & $K$ & $M$ & $\overline{M}$ & $L$ & $\overline{L}$ \\
\addlinespace[1.5pt]
$\chi_{25}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $L$ & $\overline{L}$ & $K$ & $\overline{K}$ & $\overline{M}$ & $M$ \\
$\chi_{26}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\overline{L}$ & $L$ & $\overline{K}$ & $K$ & $M$ & $\overline{M}$ \\
$\chi_{27}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $M$ & $\overline{M}$ & $L$ & $\overline{L}$ & $K$ & $\overline{K}$ \\
$\chi_{28}$ & $11904$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\overline{M}$ & $M$ & $\overline{L}$ & $L$ & $\overline{K}$ & $K$ \\
$\chi_{29}$ & $12400$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\
$\chi_{30}$ & $12896$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{31}$ & $12896$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{32}$ & $14976$ & $F$ & $\overline{F}$ & $\overline{J}$ & $J$ & $\overline{I}$ & $I$ & $\overline{H}$ & $H$ & $\overline{G}$ & $G$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{33}$ & $14976$ & $\overline{F}$ & $F$ & $J$ & $\overline{J}$ & $I$ & $\overline{I}$ & $H$ & $\overline{H}$ & $G$ & $\overline{G}$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{34}$ & $14976$ & $G$ & $\overline{G}$ & $F$ & $\overline{F}$ & $\overline{J}$ & $J$ & $\overline{I}$ & $I$ & $\overline{H}$ & $H$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{35}$ & $14976$ & $\overline{G}$ & $G$ & $\overline{F}$ & $F$ & $J$ & $\overline{J}$ & $I$ & $\overline{I}$ & $H$ & $\overline{H}$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{36}$ & $14976$ & $H$ & $\overline{H}$ & $G$ & $\overline{G}$ & $F$ & $\overline{F}$ & $\overline{J}$ & $J$ & $\overline{I}$ & $I$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{37}$ & $14976$ & $\overline{H}$ & $H$ & $\overline{G}$ & $G$ & $\overline{F}$ & $F$ & $J$ & $\overline{J}$ & $I$ & $\overline{I}$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{38}$ & $14976$ & $I$ & $\overline{I}$ & $H$ & $\overline{H}$ & $G$ & $\overline{G}$ & $F$ & $\overline{F}$ & $\overline{J}$ & $J$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{39}$ & $14976$ & $\overline{I}$ & $I$ & $\overline{H}$ & $H$ & $\overline{G}$ & $G$ & $\overline{F}$ & $F$ & $J$ & $\overline{J}$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{40}$ & $14976$ & $J$ & $\overline{J}$ & $I$ & $\overline{I}$ & $H$ & $\overline{H}$ & $G$ & $\overline{G}$ & $F$ & $\overline{F}$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{41}$ & $14976$ & $\overline{J}$ & $J$ & $\overline{I}$ & $I$ & $\overline{H}$ & $H$ & $\overline{G}$ & $G$ & $\overline{F}$ & $F$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{42}$ & $15625$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
$\chi_{43}$ & $16120$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{44}$ & $16120$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{45}$ & $19344$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{46}$ & $19344$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{47}$ & $19344$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
$\chi_{48}$ & $19344$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\addlinespace[1.5pt]
$\chi_{49}$ & $24180$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ \\
\bottomrule
\end{tabular}\par
\endgroup
\medskip
\begingroup\footnotesize
\noindent Bars denote complex conjugation. The noninteger values use the following notation:
\[
\begin{gathered}
\zeta_n=e^{2\pi i/n},\qquad
\eta_{n,r}=\sum_{k=0}^{d_n-1}\zeta_n^{r5^k},\qquad
d_{13}=d_{39}=4,\quad d_{31}=3,\\[2pt]
A=\frac{3-5\sqrt5}{2},\qquad A'=\frac{3+5\sqrt5}{2},\qquad E=-i\sqrt6,\\[2pt]
(B,C,D)=(-\eta_{13,1},-\eta_{13,4},-\eta_{13,2}),\\[2pt]
(F,G,H,I,J)=(\eta_{31,17},\eta_{31,16},\eta_{31,6},\eta_{31,2},\eta_{31,11}),\\[2pt]
(K,L,M)=(-\eta_{39,23},-\eta_{39,4},-\eta_{39,14}).
\end{gathered}
\]
\endgroup


\renewcommand{\bibliofont}{\small}
\begin{thebibliography}{NTVW12}

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S.~H.~Alavi, A.~Daneshkhah, H.~P.~Tong-Viet and T.~P.~Wakefield,
On Huppert's conjecture for the Conway and Fischer families of sporadic simple groups,
\emph{J. Aust. Math. Soc.} \textbf{94} (2013), no.~3, 289--303.

\bibitem[BTZ17]{BTZ2017}
C.~Bessenrodt, H.~P.~Tong-Viet and J.~Zhang,
Huppert's conjecture for alternating groups,
\emph{J. Algebra} \textbf{470} (2017), 353--378.

\bibitem[Bre25]{CTblLibL45}
T.~Breuer,
\emph{The GAP Character Table Library, Version 1.3.11},
GAP package, May 2025,
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\bibitem[Hup00]{HuppertI}
B.~Huppert,
Some simple groups which are determined by the set of their character degrees.~I,
\emph{Illinois J. Math.} \textbf{44} (2000), no.~4, 828--842.

\bibitem[Isa76]{Isaacs}
I.~M.~Isaacs,
\emph{Character Theory of Finite Groups},
Pure and Applied Mathematics, vol.~69, Academic Press, 1976.

\bibitem[NTW12]{NTW2012}
H.~N.~Nguyen, H.~P.~Tong-Viet and T.~P.~Wakefield,
Projective special linear groups $\PSL_4(q)$ are determined by the set of their character degrees,
\emph{J. Algebra Appl.} \textbf{11} (2012), no.~6, 1250108, 26~pp.

\end{thebibliography}

\end{document}