Subgroup ($H$) information
| Description: | $C_3^2\times A_6$ |
| Order: | \(3240\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5 \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrr}
7 & -1 & -1 & -1 \\
2 & 1 & -1 & -1 \\
-1 & -1 & 7 & -1 \\
-1 & -1 & 6 & 1
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
-1 & 3 & -1 & -1 \\
1 & -1 & -1 & -1 \\
-1 & 0 & -1 & 7 \\
2 & -1 & 5 & -1
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
4 & -1 & -1 & -1 \\
-1 & 4 & -1 & -1 \\
0 & -1 & 4 & -1 \\
-1 & 4 & -1 & 4
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
4 & -1 & -1 & -1 \\
-1 & 4 & -1 & -1 \\
6 & -1 & 4 & -1 \\
-1 & 2 & -1 & 4
\end{array}\right) \right]$
|
| Derived length: | $1$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $\PSOPlus(4,9)$ |
| Order: | \(259200\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6^2.D_4$, of order \(4147200\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_6.C_2\times \GL(2,3)$ |
| $W$ | $A_6:C_8$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $20$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\PSOPlus(4,9)$ |