Subgroup ($H$) information
| Description: | $A_6$ |
| Order: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Index: | \(18144\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrrrr}
0 & 1 & 0 & 0 & 0 & 2 \\
2 & 0 & 1 & 0 & 2 & 2 \\
1 & 1 & 0 & 0 & 1 & 1 \\
2 & 1 & 1 & 1 & 2 & 0 \\
2 & 1 & 2 & 0 & 2 & 1 \\
0 & 2 & 0 & 1 & 1 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
1 & 1 & 1 & 1 & 0 & 1 \\
1 & 2 & 0 & 1 & 1 & 0 \\
2 & 2 & 1 & 2 & 2 & 0 \\
2 & 0 & 1 & 1 & 2 & 1 \\
1 & 2 & 1 & 2 & 2 & 1 \\
1 & 1 & 0 & 1 & 1 & 1
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $\PSOMinus(6,3)$ |
| Order: | \(6531840\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5 \cdot 7 \) |
| Exponent: | \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGammaU(4,3)$, of order \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_6$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $C_2\times S_6$ | ||||
| Normal closure: | $\PSU(4,3)$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_3^4:A_6$ | $S_6$ | $C_2\times A_6$ | $S_6$ | |
| Maximal under-subgroups: | $A_5$ | $A_5$ | $C_3^2:C_4$ | $S_4$ | $S_4$ |
Other information
| Number of subgroups in this conjugacy class | $4536$ |
| Möbius function | not computed |
| Projective image | $\PSOMinus(6,3)$ |