Subgroup ($H$) information
| Description: | $C_{41}:C_5$ |
| Order: | \(205\)\(\medspace = 5 \cdot 41 \) |
| Index: | \(260702208000\)\(\medspace = 2^{20} \cdot 3^{2} \cdot 5^{3} \cdot 13 \cdot 17 \) |
| Exponent: | \(205\)\(\medspace = 5 \cdot 41 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrrr}
1 & 3 & 8 & 12 & 4 \\
-1 & 5 & 11 & 7 & 3 \\
6 & 11 & 3 & 0 & 3 \\
10 & 13 & 0 & 8 & 7 \\
12 & 3 & 9 & 8 & 14
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrr}
7 & 10 & 0 & 14 & 1 \\
11 & 8 & 5 & -1 & 4 \\
5 & 4 & 6 & 12 & 8 \\
2 & 1 & 1 & 1 & 14 \\
0 & 13 & 6 & 8 & 8
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.
Ambient group ($G$) information
| Description: | $\PSU(5,4)$ |
| Order: | \(53443952640000\)\(\medspace = 2^{20} \cdot 3^{2} \cdot 5^{4} \cdot 13 \cdot 17 \cdot 41 \) |
| Exponent: | \(1087320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 41 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(1068879052800000\)\(\medspace = 2^{22} \cdot 3^{2} \cdot 5^{5} \cdot 13 \cdot 17 \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $F_{41}$, of order \(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_{41}:C_5$ |
| Normal closure: | $\PSU(5,4)$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $260702208000$ |
| Möbius function | not computed |
| Projective image | not computed |