Subgroup ($H$) information
| Description: | $C_{12}.D_6$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,7)(2,33)(3,5)(4,65)(6,8)(9,44)(10,80)(11,42)(13,47)(14,28)(15,25)(16,59) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_3^3:S_3.C_2^4:S_5$ |
| Order: | \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.C_2^4:S_5$, of order \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_6^2:(D_6\times \GL(2,3))$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $W$ | $C_6^2:\GL(2,3)$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $90$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3.C_2^4:S_5$ |