Subgroup ($H$) information
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,9,2)(3,6,7)(4,8,5), (1,4)(2,8)(3,7)(5,9), (1,5)(2,4)(3,6)(8,9)(10,12)(11,15)(13,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_3^4:(D_4\times D_6)$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3:S_3\times \He_3).D_4^2$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $216$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:(D_4\times D_6)$ |