Subgroup ($H$) information
| Description: | $C_3\times S_3^2$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(7,8,9)(10,12,11), (2,4)(3,6)(7,8,9)(10,11,12)(13,14,15), (1,3,6)(2,5,4) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^2:C_6\times A_6$ |
| Order: | \(19440\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3.S_3^2.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $S_3^2:C_2^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $S_3^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $180$ |
| Möbius function | $-2$ |
| Projective image | $C_3^2:C_6\times A_6$ |