Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(4,5,7), (8,15)(9,10)(11,13)(12,14)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $D_6.S_4^2$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times A_4^2.C_2^2\times S_3$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-288$ |
| Projective image | $S_3\times S_4^2$ |