Subgroup ($H$) information
| Description: | $C_2^3.D_6^2$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,4,6), (9,12)(10,11), (7,14)(8,13)(9,12)(10,11), (7,9)(8,11)(10,13)(12,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and rational.
Ambient group ($G$) information
| Description: | $(C_2\times C_6^2):\GL(2,\mathbb{Z}/4)$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_3^2.A_4^2.C_2^4.C_2$ |
| $\operatorname{Aut}(H)$ | $C_6^2.(C_2^4\times A_4).C_2^3$ |
| $W$ | $D_6^2:D_4$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_6^2:\GL(2,\mathbb{Z}/4)$ |