Subgroup ($H$) information
| Description: | $C_2\times A_4\times D_6$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(6,7)(8,9), (1,2)(3,5)(12,13), (6,7)(12,13), (3,4,5)(10,11), (8,9), (6,8,13)(7,9,12), (10,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2^4:\GL(2,4)$ |
| Order: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times S_4\times S_5$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_4^2$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $S_4\times S_3^2$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $S_3\times A_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4\times A_5$ |