Subgroup ($H$) information
| Description: | $C_2\times C_4^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, b^{9}, c^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_{12}^2:C_2$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^4:C_3.D_4\times S_3$ |
| $\operatorname{Aut}(H)$ | $C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2\wr S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3\times S_3$ |