Subgroup ($H$) information
| Description: | $C_3\times C_9$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(18816\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\langle(1,5,3,9,8,7,4,2,6)(10,14,12,11,16,15,18,13,17), (10,18,11)(12,17,15)(13,16,14), (1,4,9)(2,8,5)(3,6,7)(10,11,18)(12,15,17)(13,14,16)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 \(1524096\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_9^2$ | |||
| Normalizer: | $D_9^2$ | |||
| Normal closure: | $\SL(2,8)^2$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $C_9^2$ | $S_3\times C_9$ | $C_3\times D_9$ | $C_3:D_9$ |
| Maximal under-subgroups: | $C_3^2$ | $C_9$ | $C_9$ |
Other information
| Number of subgroups in this conjugacy class | $1568$ |
| Möbius function | $0$ |
| Projective image | $\SOPlus(4,8)$ |