Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(1020\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,17)(2,3)(5,9)(7,8)(10,15)(11,12), (1,3)(2,17)(4,14)(5,9)(6,13)(7,8)(10,11)(12,15), (1,2)(3,17)(4,13)(5,7)(6,14)(8,9)(10,15)(11,12)\rangle$
|
| 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 rational.
Ambient group ($G$) information
| Description: | $\SOMinus(4,4)$ |
| Order: | \(8160\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 17 \) |
| Exponent: | \(1020\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,16).C_4$, of order \(16320\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^3$ | |
| Normalizer: | $C_2^3:A_4$ | |
| Normal closure: | $\SOMinus(4,4)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_2\times A_4$ | $C_2\times D_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ |
Other information
| Number of subgroups in this conjugacy class | $85$ |
| Möbius function | $0$ |
| Projective image | $\SOMinus(4,4)$ |