Subgroup ($H$) information
| Description: | $A_7$ |
| Order: | \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
13 & 1 & 21 \\
4 & -1 & 8 \\
0 & 20 & 8
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
16 & 13 & 14 \\
9 & 19 & 1 \\
6 & 6 & 14
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is maximal, nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 \(756000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $W$ | $A_7$, of order \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $50$ |
| Möbius function | $-1$ |
| Projective image | $\PSU(3,5)$ |