Subgroup ($H$) information
| Description: | $C_2^3.C_6^2$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,4)(2,3), (1,3)(2,4), (8,9)(10,13)(11,12)(14,15), (9,12)(13,15), (8,11)(9,12)(10,14)(13,15), (2,3,4)(5,6,7), (5,6,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $(C_6\times D_4):S_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times A_4).(C_6\times D_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(2,\mathbb{Z}/4):D_6$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $\GL(2,\mathbb{Z}/4):D_6$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $2$ |
| Projective image | $C_2^2\times C_6:S_4$ |