Subgroup ($H$) information
| Description: | $C_2^4:D_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,3,4,8)(2,5,6,7), (2,5)(6,7)(9,11)(10,12)(14,15), (1,3)(2,5)(4,8)(6,7)(10,12), (13,14,15), (1,4)(2,6)(3,8)(5,7), (9,11)(10,12), (2,6)(5,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $(D_6\times C_2^4):C_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^7.C_2^6.C_2^2)$ |
| $\operatorname{Aut}(H)$ | $(C_2^4\times D_{12}).D_4^2$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| $\card{W}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | not computed |