Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ad$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Ambient group ($G$) information
| Description: | $C_6^2:S_3^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8^2.D_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{12}:S_3$ | ||||
| Normalizer: | $C_{12}:D_6$ | ||||
| Normal closure: | $C_3^2:C_4$ | ||||
| Core: | $C_2$ | ||||
| Minimal over-subgroups: | $C_3:C_4$ | $C_{12}$ | $D_4$ | $C_2\times C_4$ | $D_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $162$ |
| Projective image | $C_3^4:C_2^3$ |