Subgroup ($H$) information
| Description: | $\GL(2,4)$ |
| Order: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,6,3)(2,4,10)(5,8,7), (1,4)(3,9)(6,7)(8,10)(11,12,13), (11,13,12)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $A_6:S_3$ |
| Order: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6:D_6$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2\times S_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $A_5$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_3$ | |||
| Normalizer: | $\GL(2,4)$ | |||
| Normal closure: | $C_3\times A_6$ | |||
| Core: | $C_3$ | |||
| Minimal over-subgroups: | $C_3\times A_6$ | |||
| Maximal under-subgroups: | $A_5$ | $C_3\times A_4$ | $C_3\times D_5$ | $C_3\times S_3$ |
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $A_6:S_3$ |