Subgroup ($H$) information
| Description: | $C_2^2:C_9$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{2}c^{2}, a^{6}, b^{3}c^{3}, c^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_6^2:C_{18}$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_6^2:S_3^2$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3\times S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_3\times S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^2$ | |
| Normalizer: | $C_3^2.A_4$ | |
| Normal closure: | $C_6^2:C_9$ | |
| Core: | $C_2\times C_6$ | |
| Minimal over-subgroups: | $C_3^2.A_4$ | |
| Maximal under-subgroups: | $C_2\times C_6$ | $C_9$ |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:C_6$ |