Subgroup ($H$) information
| Description: | $C_6\times S_3^2$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(3,5)(4,6), (1,3,5)(7,14)(8,13)(9,12)(10,11), (7,11,9)(10,12,14), (1,3,5)(2,6,4), (7,14)(8,13)(9,12)(10,11), (1,6)(2,3)(4,5)(7,14)(9,12)(10,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_2\times D_6^2):A_4$ |
| Order: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.(C_2^4\times A_4).C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_6^2:C_2^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\card{W}$ | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $16$ |
| Möbius function | not computed |
| Projective image | not computed |