Subgroup ($H$) information
| Description: | $C_7:C_3$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(6000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{3} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
21 & 20 & 23 \\
0 & 4 & 3 \\
7 & 17 & 12
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
11 & 6 & 18 \\
6 & 10 & 1 \\
21 & 18 & 20
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
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 \(756000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $W$ | $C_7:C_3$, of order \(21\)\(\medspace = 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $C_7:C_3$ | ||
| Normal closure: | $\PSU(3,5)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\PSL(2,7)$ | $\PSL(2,7)$ | $\PSL(2,7)$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $6000$ |
| Möbius function | $-1$ |
| Projective image | $\PSU(3,5)$ |