Subgroup ($H$) information
| Description: | $D_6\times S_4$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(11,14)(12,13), (2,4,3)(5,6,8), (2,4)(3,9), (2,9)(3,4), (2,3,9)(11,14)(12,13), (3,4)(5,6)(11,13)(12,14), (1,7)(2,3)(11,14)(12,13)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $S_5^2:D_4$ |
| Order: | \(115200\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_5^2).D_4^2$, of order \(460800\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^2:D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_2^3:D_6^2$ | ||
| Normal closure: | $A_5^2:C_2^3$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $D_6\times S_5$ | $C_2^2:D_6^2$ | $C_2^2:D_6^2$ |
Other information
| Number of subgroups in this autjugacy class | $400$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $S_5^2:D_4$ |