Subgroup ($H$) information
| Description: | $C_4^2:C_2^3$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,4,3,8)(2,7,6,5), (1,8,3,4)(2,7,6,5), (1,2)(3,6)(4,7)(5,8)(9,10)(11,12) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{12}.(D_4\times S_4)$, of order \(786432\)\(\medspace = 2^{18} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $D_4^3.D_4$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\card{W}$ | \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | not computed |