Subgroup ($H$) information
| Description: | $C_2^2\times C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ae^{9}, bcde^{6}, d^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $(C_2^2\times D_4):D_6$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $8$ |
| Projective image | $C_2^3\times D_6$ |