Subgroup ($H$) information
| Description: | $C_3^2\times D_6$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,3,2)(4,5)(7,15)(8,10)(9,14)(11,13), (4,6,5), (1,2,3)(4,5)(7,8,11)(10,13,15), (4,5), (1,3,2)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3\times S_3\times {}^2G(2,3)$ |
| Order: | \(27216\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^2\times {}^2G(2,3)$, of order \(54432\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $D_6\times \GL(2,3)$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $252$ |
| Möbius function | $2$ |
| Projective image | $S_3\times {}^2G(2,3)$ |