Subgroup ($H$) information
| Description: | $C_3:S_4$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(1750\)\(\medspace = 2 \cdot 5^{3} \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
2 & 7 & 0 \\
4 & 6 & 11 \\
6 & 20 & 10
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
0 & 7 & 10 \\
5 & -1 & 19 \\
22 & 9 & 8
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
11 & 3 & 4 \\
0 & 15 & 14 \\
23 & 4 & 22
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
16 & 23 & 18 \\
5 & 11 & 11 \\
14 & 3 & 13
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
6 & 23 & 20 \\
14 & 8 & 11 \\
14 & 14 & 22
\end{array}\right) \right]$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
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 \(756000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $C_3:S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | |||||
| Normalizer: | $C_3:S_4$ | |||||
| Normal closure: | $\PSU(3,5)$ | |||||
| Core: | $C_1$ | |||||
| Minimal over-subgroups: | $A_7$ | $A_7$ | $A_7$ | |||
| Maximal under-subgroups: | $C_3\times A_4$ | $C_3:D_4$ | $S_4$ | $S_4$ | $S_4$ | $C_3:S_3$ |
Other information
| Number of subgroups in this conjugacy class | $1750$ |
| Möbius function | $2$ |
| Projective image | $\PSU(3,5)$ |