Subgroup ($H$) information
| Description: | $C_4.D_4^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b^{3}c^{5}d^{3}e^{4}, c^{6}d^{4}, c^{2}d^{6}e^{7}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_4^3.(C_2^2\times S_4)$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_8^2.C_6.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^6.C_2^6.C_2$ |
| $W$ | $C_2\times D_4^2$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_8^2:(S_3\times D_4)$ |