Subgroup ($H$) information
| Description: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, b^{3}, c^{3}d^{3}$
|
| Nilpotency class: | $2$ |
| 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: | $C_6^2:D_{12}$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8.C_2^2.D_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(S)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_4:D_4$ | |||
| Normal closure: | $C_6^2:D_{12}$ | |||
| Core: | $C_4$ | |||
| Minimal over-subgroups: | $C_2\times D_{12}$ | $C_4:D_4$ | ||
| Maximal under-subgroups: | $C_2^3$ | $C_2\times C_4$ | $D_4$ | $D_4$ |
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:D_6$ |