Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(72576\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(7\) |
| Generators: |
$\langle(10,15,16,17,11,12,13)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $\SOPlus(4,8)$ |
| Order: | \(508032\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,8)^2:C_6$, of order \(1524096\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $\GL(2,8)$ | |||||
| Normalizer: | $D_7.\SL(2,8)$ | |||||
| Normal closure: | $\SL(2,8)^2$ | |||||
| Core: | $C_1$ | |||||
| Minimal over-subgroups: | $F_8$ | $C_7^2$ | $C_{21}$ | $C_{14}$ | $D_7$ | $D_7$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $72$ |
| Möbius function | $0$ |
| Projective image | $\SOPlus(4,8)$ |