Subgroup ($H$) information
| Description: | $C_2^5$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(5,6)(7,8)(9,10)(11,12), (1,2)(3,4)(13,14)(15,16), (3,4)(5,8)(6,7)(9,11)(10,12)(15,16), (5,6)(7,8)(13,14)(15,16), (13,14)(15,16)\rangle$
|
| 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: | $(D_4\times C_2^3).Q_{16}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The ambient group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5:C_4.D_4^2$, of order \(8192\)\(\medspace = 2^{13} \) |
| $\operatorname{Aut}(H)$ | $\GL(5,2)$, of order \(9999360\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $\operatorname{res}(S)$ | $D_4:D_4$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_4:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $(C_2^4:C_4) . (C_2\times C_4)$ |