Computing the conjugacy classes and character table of a non-split extension 2⁶·(2⁵:S₆) from a split extension 2⁶:(2⁵:S₆)

1.   Introduction

In the papers ^[2] and ^[16], the character tables of the non-split extension $\overline{G_3} = 2^6{{}^{\cdot}}Sp_6(2)$ and split extension $2^6{:}Sp_6(2)$ were successfully computed by the method of Fischer-Clifford matrices ^[8]. In the ATLAS ^[6] we found that $N_{Fi_{22}}(2^6)\cong 2^6{:}Sp_6(2)$ is a maximal subetaoup of the smallest Fischer sporadic simple group $Fi_{22}$ of index 694980. Here the elementary abelian group $2^6$ is a pure $2B$-group, where $2B$ denotes a class of involutions in $Fi_{22}$. Dempwolff in ^[7] proved that a unique non-split extension (up to isomorphism) of the form $\overline{G_n} = 2^{2n}{{}^{\cdot}}Sp_{2n}(2)$ does exist for all $n \geq 2$, where $\overline{G_n}/2^{2n}\cong Sp_{2n}(2)$ acts faithfully on $2^{2n}$. In ^[2] it was noted that $2^6{:}Sp_6(2)$ and $\overline{G_3}$ give rise to the same character table. The groups $2^6{:}Sp_6(2)$ and $\overline{G_3}$ have subetaoups of types $M_{1} = 2^6{:}(2^5{:}S_6)$ and $\overline{G} = 2^6{{}^{\cdot}}(2^5{:}S_6)$, where $M_1$ and $\overline{G}$ are pre-images of a maximal subetaoup $2^5{:}S_6$ of index 63 in $Sp_6(2)$, under the natural epimorphism modulo $2^6$.

In this paper, it will be shown that the character tables of $M_1$ and $\overline{G}$ coincide and how the conjugacy classes of $\overline{G}$ can be obtained from the classes of $M_1$ by "restricting" characters of $\overline{G_3}$ (see Section 5 of this paper) to characters of $M_1$. In this regard, the format of the character tables of $M_1$ and $\overline{G}_3$ (see ^[2] and ^[13]) which were obtained by the method of Fischer-Clifford matrices, plays an important role. The power maps of $\overline{G}$ and the fusion map of $\overline{G}$ into $\overline{G}_3$ are also computed. Most of our computations are done in the computer algebra systems MAGMA ^[5] and GAP ^[21]. For concepts and definitions used in this paper, the readers are referred to the review paper on Fischer-Clifford theory ^[3] and ^{[1,11,12,17,18,19]}.

2.   Theory of Fischer-Clifford matrices

Since the ordinary character tables of the groups $2^6{:}Sp_6(2)$, $\overline{G_3}$, $M_1$ and $\overline{G}$ have been computed by the technique of Fischer-Clifford matrices, a brief theoretical background of this technique will be given in this section. In Section 4, it will be shown that only the ordinary irreducible characters of the inertia factors will be used in the construction of the character table of $\overline{G}$. Therefore, only the case where every irreducible character of $N$ can be extended to its inertia group in the extension group $N{.}G$ will be discussed. Here the author will follow closely the work of the authors in ^[16].

Let $\overline{G} = N{.}G$ be an extension of $N$ by $G$ and $\theta\in Irr(N)$, where $Irr(N)$ denotes the irreducible characters of $N$. Define $\theta^{g}$ by $\theta^{g}(n) = \theta({gng^{-1}})$ for $g\in \overline{G}$, $n\in N$ and $\theta^{g}\in Irr(N)$. Let $\overline{H} = \left\{x\in \overline{G} | \theta^{x} = \theta\right\} = I_{\overline{G}}(\theta)$ be the inertia group of $\theta$ in $\overline{G}$. We say that $\theta$ is extendible to $\overline{H}$ if there exists $\phi \in Irr(\overline{H})$ such that $\phi\downarrow_{N} = \theta$. If $\theta$ is extendible to $\overline{H}$, then by Gallagher ^[11], we have

Let $\overline{G}$ have the property that every irreducible character of $N$ can be extended to its inertia group. Now let $\theta_1 = 1_N, \theta_2, \cdots, \theta_t$ be representatives of the orbits of $\overline{G}$ on $Irr(N)$, $\overline{H}_i = I_{\overline{G}} (\phi_i)$, $1 \le i \le t$, $\phi_i \in Irr (\overline{H}_i)$ be an extension of $\theta_i$ to $\overline{H}_i$ and $\beta \in Irr (\overline{H}_i)$ such that $N \subseteq ker (\beta)$. Then

Hence the irreducible characters of $\overline{G}$ will be divided into blocks, where each block corresponds to an inertia group $\overline{H}_i$.

Let $H_i$ be the inertia factor group and $\phi_i$ be an extension of $\theta_i$ to $\overline{H}_i$. Take $\theta_1 = 1_N$ as the identity character of $N$, then $\overline{H}_1 = \overline{G}$ and $H_1\cong G$. Let $X (g) = \{ x_1, x_2, \cdots, x_{c(g)} \}$ be a set of representatives of the conjugacy classes of $\overline{G}$ from the coset $N \overline{g}$ whose images under the natural homomorphism $\overline{G} \longrightarrow G$ are in the class $[g]$ of $G$ and we take $x_1 = \bar{g}$. We define

where $y_k$ runs over representatives of the conjugacy classes of elements of $H_i$ which fuse into $[g]$. Let $\left\{y_{l_k}\right\}$ be the representatives of conjugacy classes of $\overline{H}_i$ which contain liftings of $y_k$ under the natural homomorphism $\overline{H_i} \longrightarrow H_i$. Then we define the Fischer-Clifford matrix $M (g)$ by $M(g) = (a_{(i, y_k)}^{j})$, where

with columns indexed by $X (g)$ and rows indexed by $R (g)$ and where $\sum_{l}^{\prime}$ is the summation over all $l$ for which $y_{l_k} \thicksim x_j$ in $\overline{G}$. Then the partial character table of $\overline{G}$ on the classes $\{ x_1, x_2, \cdots, x_{c (g)} \}$ is given by $\left[\begin{array}{c} C_1 (g) \, M_1 (g) \\ C_2 (g) \, M_2 (g) \\ \vdots \\ C_t (g) \, M_t (g) \end{array} \right]$ where the Fischer-Clifford matrix $M (g) = \left[\begin{array}{c} M_1 (g) \\ M_2 (g) \\ \vdots \\ M_t (g) \end{array} \right] \,$ is divided into blocks $M_i(g)$ with each block corresponding to an inertia group $\overline{H}_i$ and $C_i (g)$ is the partial character table of ${H_i}$ consisting of the columns corresponding to the classes that fuse into $[g]$. Hence the full character table of $\overline{G}$ will be $\left[\begin{array}{c} \Delta_1 \\ \Delta_2 \\ \vdots \\ \Delta_t \end{array} \right]$, where $\Delta_i$ = $[C_i(1)M_i(1)|C_i(g_2)M_i(g_2)|...|C_i(g_k)M_i(g_k)]$ with $\{1, g_1, g_2, ..., g_k\}$ the representatives of conjugacy classes of $G$. We can also observe that $|Irr(\overline{G})|$ = $|Irr(H_1)|$ + $|Irr(H_2)|$ +...+ $|Irr(H_t)|$.

3.   The action of $G$ on $N$ and $Irr(N)$

The group $\overline{G_3} = 2^6{{}^{\cdot}}Sp_6(2)$ was constructed in ^[2] as a permutation group on 128 points and it was shown that $\overline{G_3}$ has an inertia group $\overline{G} = 2^6{{}^{\cdot}}(2^5{:}S_6)$ which belongs to a non-split extension of a reducible module of dimension 6 over $GF(2)$ for the maximal subetaoup $2^5{:}S_6$ of $Sp_6(2)$. Using the generators of $\overline{G_3}$ given in ^[2], the group $\overline{G}$ is constructed as the centralizer $C_{\overline{G}_3}(2A)$ of the class of involutions $2A$ within $\overline{G_3}$.

Let $\overline{G} = 2^6{{}^{\cdot}}(2^5{:}S_6)$ be the non-split extension of $N = 2^6$ by $G = 2^5{:}S_6$. The group $2^5{:}S_6$ is the stabilizer of a vector in the action of $Sp_6(2)$ on its natural 6-dimensional module $2^6$. This action is the same in both the split and non-split extensions $2^6{:}Sp_6(2)$ and $\overline{G_3}$. This immediately defines the action of $2^5{:}S_6$ on the module $2^6$. Note that the action of the split extension $M_1 = 2^6{:}(2^5{:}S_6)$ on $2^6$ is the same as the action of $2^5{:}S_6$ on $2^6$. The group $G$ can be constructed as a matrix group of dimension 6 over the finite field $GF(2)$ within $Sp_{6}(2)$. Now with the action of $G$ on $N = 2^6$, where we view $N$ as the vector space of dimension six over $GF(2)$, we will obtain four orbits of lengths 1, 1, 30 and 32 with corresponding point stabilizers $G$, $G$, $2^4{:}S_5$ and $S_6$, respectively. By Brauer's Theorem ^[10] the action of $G$ on $Irr(N)$ will also produce 4 orbits and since the action is self-dual, the orbit lengths will be $1, 1, 30$ and $32$ with corresponding inertia factor groups $H_1 = H_2 = G$, $H_3 = 2^4{:}S_5$ and $H_4 = S_6$.

4.   The ordinary character table of $\overline{G}=2^6{{}^{\cdot}}(2^5{:}S_6)$

Having obtained the inertia factors $H_1 = H_2 = G$, $H_3 = 2^4{:}S_5$ and $H_4 = S_6$ for the action of $G$ on $Irr(N)$, we can formed the Fischer-Clifford matrix $M(1A)$ corresponding to the identity coset $N1_{\overline{G}} = N$ as follows:

The column weights above the matrix $M(1A)$ are the centralizer orders $|C_{\overline{G}}(\overline{g})|$ of the classes $1A, 2A, 2B$ and $2C$ of $\overline{G}$ (see Table 5) coming from the identity coset $N(\overline{1A}) = N$ by means of the technique of coset-analysis (see ^[14], ^[15] and ^[16]). Whereas, the row weights to the left of the matrix $M(1A)$ represent the centralizer orders $|C_{H_i}(1A)|$ of the inertia factors $H_i$ on the identity element $1A$.

Table 1 is the partial ordinary character table of $\overline{G}$ on the classes $1A, 2A, 2B$ and $2C$ of $\overline{G}$, where each of the 4 lines of Table 1 corresponds to the first row of entries of the sub-matrices $C_i(1A)M_i(1A), i = 1, 2, 3, 4$. $M_i(1A)$ and $C_i(1A)$ correspond to the rows of the Fischer-Clifford matrix $M(1A)$ and columns of the projective character tables of the inertia factors $H_i$, respectively, which are associated with the classes $[1A]_{H_i}$ of the inertia factors $H_i$ which fuse into the class $[1A]_{G}$ of $G$. Also, note that the character values in the 1st column of Table 1 are the degrees of the ordinary irreducible characters $\chi_1, \chi_{38}, \chi_{38+t_2}$ and $\chi_{38+t_2+t_3}$ of $\overline{G}$. The characters $\chi_1, \chi_{38}, \chi_{38+t_2}$ and $\chi_{38+t_2+t_3}$ occupy the first position for each block of characters coming from an inertia subetaoup $\overline{H}_i$ of $\overline{G}$, where $37, t_2$ and $t_3$ represent the number $|IrrProj(H_i, \alpha_i)|$ of irreducible projective characters with associated factor set $\alpha_i$ for the inertia factors $H_1, H_2$ and $H_3$, respectively. Now $deg(\eta_1) = 1, deg(\phi_1) = a, deg(\psi_1) = b$ and $deg(\gamma_1) = c$ are the degrees of the irreducible projective characters ${\bf 1}_G$, $\phi_1$, $\psi_1$ and $\gamma_1$ which occupy the first position in each set $IrrProj(H_i, \alpha_i)$, $i = 1, 2, ..., 4$, respectively.

We copy a small part of the ordinary character table of $2^6{{}^{\cdot}}Sp_6(2)$ (see Table 11.12 in ^[4]), containing the values of the character $63a$ on the classes $1A, 2A$:

Now the classes $1A, 2A, 2B$ and $2C$ of $\overline{G}$ consist of the elements of $N$ (see Table 5). If we decompose $(63a)_N$ into the the set $Irr(N)$ and also notice that $< (63a)_{N}, 1_N >$ = 0, then $(63a)_N = a(\chi_{38})_N+b(\chi_{38+t_2})_N+ c(\chi_{38+t_2+t_3})_N$, where $a, b$ and $c$ are defined as above. If we take into account the fusion of the classes $1A, 2A, 2B$ and $2C$ of $\overline{G}$ into the classes of $1A$ and $2A$ of $\overline{G_3}$, and the decomposition of $(63a)_N$ into the set $Irr(N)$, then the following set of equations (by restricting the character values of $63a$ to Table 1) is obtained:

1. $(63a)_N(1A) = a+30b+32c = 63$

2. $(63a)_N(2A) = a+30b-32c = -1$

3. $(63a)_N(2B) = a-2b = -1$

4. $(63a)_N(2C) = -a = -1$

Solving the above equations simultaneously, we obtain that $a = b = 1 = c = 1$ and hence $deg(\phi_1) = deg(\psi_1) = deg(\gamma_1) = 1$. We can conclude that only the ordinary irreducible characters tables of the inertia factors $H_i$ will be involved in the construction of the ordinary character table of $2^6{{}^{\cdot}}(2^5{:}S_6)$. This means that the ordinary irreducible characters of the split extension $M_1 = 2^6{:}(2^5{:}S_6)$ (Table 9.7 in ^[13]) are the same as the ones for the non-split extension $\overline{G}$, but the class orders of the two groups will differ as it will be shown in Section 5.

5.   The conjugacy classes of $\overline{G}$

In this section, we will compute the order of an element $\overline{g}$ in a conjugacy class $[\overline{g}]_{\overline{G}}$ of $\overline{G}$ from the conjugacy classes and ordinary irreducible characters of both $M_1$ and $\overline{G}_3$. For both $\overline{G}$ and $M_1$, the centralizer orders for each class of elements coming from a corresponding coset $N\overline{g}$, $g \in 2^5{:}S_6$, will be the same, but their class orders may be different. The method of coset-analysis was used to compute the conjugacy classes of elements of $2^6{:}(2^{5}{:} S_{6})$ (see Table 5) and $\overline{G}_3$ (Table 1 in ^[2]). Let $\overline{G} = N{.} G$ be an extension of $N$ by $G$, where $N$ is abelian. Then for $g\in G$, we write $\overline{g}$ for a lifting of g in $\overline{G}$ under the natural homomorphism $\overline{G}\longrightarrow G$. We consider a coset $N\overline{g}$ for each class representative $g$ of $G$, writing $k$ for number of orbits of $N$ acting by conjugation on the coset $N\overline{g}$, and $f_j$ for the numbers of these fused by the action of $\left\{\overline{h}: h \in C_G(g)\right\}$. Note if $\overline{G}$ is a split extension then $\overline{g}$ becomes $g$. The order of the centralizer $C_{\overline{G}}(x)$ for each element $x \in \overline{G}$ in a conjugacy class $[x]_{\overline{G}}$ is given by $|C_{\overline{G}}(x)| = \frac{k|C_G(g)|}{f_j}$.

For example, let consider the classes of $M_1 = 2^6{:}(2^5{:}S_6)$ obtained from the cosets $N(2A)$ and $N(2E)$ (see Table 3), where $2A$ and $2E$ are classes of involutions in $2^5{:}S_6$. In addition, we consider also the partial character table of $M_1$ corresponding to the cosets $N(2A)$ and $N(2E)$ (see ^[13]), which was computed by the technique of Fischer-Clifford matrices. We obtained also from ^[13] that $(\phi_{1} = 63a)_{2^6{:}(2^5{:}S_6)}$ = $\chi_{38} + \chi_{75} + \chi_{127}$, $(\phi_2 = 63b)_{2^6{:}(2^5{:}S_6)}$ = $\chi_{39} + \chi_{76} + \chi_{128}$, $(\phi_3 = 315a)_{2^6{:}(2^5{:}S_6)}$ = $\chi_{43} + \chi_{86} + \chi_{94} + \chi_{132}$ and $(\phi_4 = 315b)_{2^6{:}(2^5{:}S_6)}$ = $\chi_{40} + \chi_{75} + \chi_{80} + \chi_{94} + \chi_{130}$, where $\phi_{1}$, $\phi_{2}$, $\phi_{3}$ and $\phi_{4}$ are ordinary irreducible characters of $2^6{:}Sp_6(2)$ of degrees 63 and 315 which are restricted to irreducible characters of $M_1$ by the technique of set intersection (see ^[9,14,16]).

From Table 4 we notice that classes $2A$ and $2E$ of $2^5{:}S_6$ are fusing into the class $2A$ of $Sp_6(2)$. Hence the classes of $\overline{G}$, which will be obtained from the cosets $N(\overline{2A})$ and $N(\overline{2E})$ using coset analysis, will fuse into the classes of $\overline{G}_3$ lying above the class $2A$ of $Sp_6(2)$. Since the character tables of $\overline{G}$ and $M_1$ coincide, the corresponding cosets $N(\overline{2A})$ and $N(\overline{2E})$ for both of the groups will produce the same number of classes and share the same class centralizer orders and partial character tables. Also $\overline{G}_3$ and $2^6{:}Sp_6(2)$ share the same character table and therefore we can expect that the irreducible character $\chi_{32}$ of degree 63 of $\overline{G}_3$ (see Table 2) will restrict to the same irreducible characters as above-mentioned character $\phi_2 = 63b$.

Suppose that Table 3 is the partial character table of $\overline{G}$ corresponding to the cosets $N$, $N(\overline{2A})$ and $N(\overline{2E})$ and irreducible characters of degrees 1, 30 and 32. Now the ordinary character $\chi_{32}$ of $\overline{G}_3$ in Table 2 will restrict to the sum of the irreducible characters $\chi_{39}$, $\chi_{76}$ and $\chi_{128}$ of $\overline{G}$ in Table 3. If the character values of $\chi_{32}$ on the classes $4A$, $4B$ and $2B$ coming from the coset $N(\overline{2A})$ in Table 2 and the character values of the restricted character $(\chi_{32})_{\overline{G}} = \chi_{39}+\chi_{76}+\chi_{128}$ on the classes $2D, 2E, 4A, 2L, 2M, 2N, 2O, 4I$ and $4J$ coming from the cosets $N(\overline{2A})$ and $N(\overline{2E})$ in Table 3 are taking into consideration, then the class orders of $2D, 2E, 2L, 2M, 2N$ and $2O$ are forced to change from order 2 to order 4 whereas the class orders of $4A, 4I$ and $4J$ are forced to change from order 4 to order 2. Hence we obtained the classes of $\overline{G}$, with their respective class orders and centralizer orders (see Table 5), associated with the cosets $N(\overline{2A})$ and $N(\overline{2E})$. In a similar fashion, we obtained all the classes of $\overline{G}$, with their class and centralizer orders, using the above restricted characters $\phi_{1}$, $\phi_{2}$, $\phi_{3}$ and $\phi_{4}$ together with the ordinary character tables of $\overline{G}_3$ and $M_1$. See Table 5 where all the information concerning the conjugacy classes of $M_1$ and $\overline{G}$ are listed. For the explanation of the parameters used in Table 5 the readers are referred to ^[16] and ^[20].

6.   The power maps of the elements of $2^6{:}(2^5{:}S_6)$

By restricting some ordinary characters of $\overline{G}_3$ to $\overline{G}$ and also computing the structure constants (using GAP) for the set $Irr(\overline{G})$, we ensure that the consistency checks of Programme E ^[22] for the set $Irr(\overline{G})$ are satisfied. The information about the conjugacy classes found in Table 5 can be used to compute the power maps for the elements of $\overline{G}$ and Programme E is used to confirm that the character table of $\overline{G}$ produces the unique p-powers listed in Table 6.

7.   The fusion of $2^6{{}^\cdot}(2^5{:}S_6)$ into $2^6{{}^\cdot}SP_6(2)$

By making use of the values of $\phi_{1}$, $\phi_{2}$, $\phi_{3}$ and $\phi_{4}$ on the classes of $2^6{{}^\cdot}Sp_6(2)$, the values of $(\phi_{1})_{2^6{{}^\cdot}(2^5{{}^\cdot}S_6)}$, $(\phi_{2})_{2^6{{}^\cdot}(2^5{:}S_6))}$, $(\phi_{3})_{2^6{{}^\cdot}(2^5{:}S_6)}$ and $(\phi_{4})_{2^6{{}^\cdot}(2^5{:}S_6)}$ on the classes of $2^6{{}^\cdot}(2^5{:}S_6)$, Table 4 and the permutation character $\chi(2^6{{}^\cdot}Sp_6(2)|2^6{{}^\cdot}(2^5{:}S_6))$ = $1a + 27a + 35a$ of degree $63$ of $2^6{{}^\cdot}Sp_6(2)$ acting on $2^6{{}^\cdot}(2^5{:}S_6)$, the complete fusion map of $2^6{{}^\cdot}(2^5{:}S_6)$ into $2^6{{}^\cdot}Sp_6(2)$ is computed and is given in Table 7.

Acknowledgments

The Department of Mathematics and Physics at Cape Peninsula University of Technology is acknowledge for the time and space provided for the author to complete the research article. I am most grateful to my Lord Jesus Christ.

Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this paper.