Subgroup ($H$) information
| Description: | $C_2^3\times D_6$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,4)(2,6)(3,8)(5,7)(9,11)(10,12), (1,3)(2,5)(4,8)(6,7), (10,12), (2,5)(6,7)(9,11)(10,12)(14,15), (13,14,15), (2,6)(5,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^7.C_2^6.C_2^2)$ |
| $\operatorname{Aut}(H)$ | $C_2^4.A_8\times S_3$, of order \(1935360\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\card{W}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $32$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $0$ |
| Projective image | not computed |