Subgroup ($H$) information
| Description: | $\GL(2,4):D_4$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Index: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,3,6)(2,5,4)(7,8,9)(10,15)(11,14)(12,13), (8,9)(10,15,12,13)(11,14), (1,5)(3,6)(7,8,9)(10,12)(13,15), (7,8)(13,15), (10,12)(13,15), (7,8,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3^3:S_4\times S_6$ |
| Order: | \(466560\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $\GL(2,4):C_2^4$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $W$ | $D_6\times S_5$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_3:D_4\times S_5$ | ||
| Normal closure: | $A_6\times C_3^3:S_4$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $A_5\times S_3^2:S_3$ | $C_3:D_4\times A_6$ | $C_3:D_4\times S_5$ |
Other information
| Number of subgroups in this autjugacy class | $324$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^3:S_4\times S_6$ |