Subgroup ($H$) information
| Description: | $C_3:S_3^2$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(7,8,9)(10,12,11), (2,3)(5,6)(7,10)(8,12)(9,11), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (4,5,6)(10,12,11), (1,2,3)(4,6,5)(7,8,9)(10,12,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^3:D_{12}$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^4:C_2$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $S_3\wr S_3$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $S_3^3:C_2$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $S_3^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $S_3^2:S_3$ | |||
| Normal closure: | $C_3^2:S_3^2$ | |||
| Core: | $C_3^2:C_6$ | |||
| Minimal over-subgroups: | $C_3^2:S_3^2$ | $S_3^2:S_3$ | ||
| Maximal under-subgroups: | $C_3^2:C_6$ | $C_3^2:C_6$ | $S_3^2$ | $S_3^2$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $C_3^3:D_{12}$ |