Subgroup ($H$) information
| Description: | $C(2,3)$ |
| Order: | \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
| Index: | \(468\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 13 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrrrr}
2 & 0 & 0 & 1 & 2 & 0 \\
2 & 0 & 1 & 1 & 0 & 1 \\
2 & 1 & 0 & 0 & 2 & 2 \\
1 & 0 & 0 & 2 & 2 & 0 \\
2 & 0 & 0 & 2 & 2 & 0 \\
0 & 1 & 2 & 1 & 1 & 0
\end{array}\right), \left(\begin{array}{rrrrrr}
0 & 1 & 0 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 & 0 & 0 \\
0 & 2 & 1 & 0 & 0 & 0 \\
2 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 2 & 0 & 0 & 2 \\
2 & 0 & 1 & 1 & 1 & 1
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $C_2\times \PSL(4,3)$ |
| Order: | \(12130560\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| Exponent: | \(4680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $117$ |
| Möbius function | not computed |
| Projective image | not computed |