Subgroup ($H$) information
| Description: | $C_5:C_8$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(3150\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
4 & -1 & -1 \\
2 & 16 & -1 \\
22 & 18 & 4
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
16 & 7 & 23 \\
7 & 10 & 15 \\
18 & 0 & 12
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
21 & 23 & 15 \\
15 & 2 & 7 \\
15 & 15 & 8
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
14 & 6 & 22 \\
23 & 16 & 14 \\
9 & 21 & 15
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 \(756000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $C_5:C_8$ | |
| Normal closure: | $\PSU(3,5)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $\He_5:C_8$ | $\SL(2,5):C_2$ |
| Maximal under-subgroups: | $C_5:C_4$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $3150$ |
| Möbius function | $1$ |
| Projective image | $\PSU(3,5)$ |