Subgroup ($H$) information
| Description: | $C_{63}$ |
| Order: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Index: | \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Generators: |
$\langle(1,8,3,7,6,4,5)(10,17,11)(12,14,13)(15,18,16), (1,8,3,7,6,4,5), (1,3,6,5,8,7,4)(10,13,18,17,12,16,11,14,15)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 \(1524096\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_6^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2016$ |
| Möbius function | $0$ |
| Projective image | $\SOPlus(4,8)$ |