Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$ad^{3}, e^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $(C_3\times C_6^3):D_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8.C_6.C_2^4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_6^2:C_2^2$ | |||
| Normalizer: | $C_6^2:C_2^2$ | |||
| Normal closure: | $C_6^2:S_3^2$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $C_2\times C_6$ | $D_6$ | $C_2^3$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $144$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $(C_3\times C_6^3):D_4$ |