Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16), (1,6)(2,5)(3,8)(4,7)(9,10)(11,12)(13,14)(15,16)\rangle$
|
| 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 rational.
Ambient group ($G$) information
| Description: | $D_4^2:C_2^3$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.D_4^3$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{W}$ | $1$ |
Related subgroups
| Centralizer: | $C_2^5$ | ||||
| Normalizer: | $C_2^5$ | ||||
| Normal closure: | $C_2\times D_4^2$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_2^4$ | $C_2^4$ | |||
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ | $C_2^2$ | $C_2^2$ | $C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $256$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | $0$ |
| Projective image | not computed |