Subgroup ($H$) information
| Description: | $A_6.C_2$ |
| Order: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Index: | \(175\)\(\medspace = 5^{2} \cdot 7 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
7 & 9 & 8 \\
10 & 8 & 13 \\
14 & 18 & 3
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
14 & 7 & 2 \\
20 & 22 & 6 \\
8 & 23 & 19
\end{array}\right) \right]$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, almost simple, and nonsolvable.
Ambient group ($G$) information
| Description: | $\PSU(3,5)$ |
| Order: | \(126000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGammaU(3,5)$, of order \(756000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_6.C_2$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $175$ |
| Möbius function | $-1$ |
| Projective image | $\PSU(3,5)$ |