Subgroup ($H$) information
| Description: | $C_2^4:D_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(8,13)(10,11), (7,13)(8,14)(9,11)(10,12), (1,3,5)(2,6,4), (9,12)(10,11) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.
Ambient group ($G$) information
| Description: | $D_6^2:C_2^2:A_4$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.(C_4\times A_4).C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^8.S_3^2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| $W$ | $D_4\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $18$ |
| Möbius function | $0$ |
| Projective image | $C_6^2.(D_4\times A_4)$ |