Subgroup ($H$) information
| Description: | $\SL(2,3)$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(5250\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
19 & 7 & 8 \\
12 & 23 & 11 \\
17 & 11 & 18
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
11 & 13 & 12 \\
21 & 6 & 17 \\
12 & 9 & 7
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
0 & 1 & 23 \\
11 & -1 & 18 \\
2 & 23 & 1
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
20 & 22 & 16 \\
2 & 3 & 2 \\
1 & 0 & 12
\end{array}\right) \right]$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable.
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 \(756000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $\GL(2,3)$ | |
| Normal closure: | $\PSU(3,5)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $\SL(2,5)$ | $\GL(2,3)$ |
| Maximal under-subgroups: | $Q_8$ | $C_6$ |
Other information
| Number of subgroups in this conjugacy class | $2625$ |
| Möbius function | $0$ |
| Projective image | $\PSU(3,5)$ |