Subgroup ($H$) information
| Description: | $C_{12}\times D_6$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(270\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2,4,3)(7,15,10)(8,13,11)(9,14,12), (1,4)(2,3)(7,11,14)(8,12,15)(9,10,13) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^2:S_3\times S_6$ |
| Order: | \(38880\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\AGL(2,3).A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^5:D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $540$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3:S_3\times S_6$ |