Subgroup ($H$) information
| Description: | $D_6\times F_5$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(11,12)(13,14), (2,3,9,10,4)(5,8,6)(11,12)(13,14), (1,7)(3,9,4,10)(11,13)(12,14), (2,9,10,3)(5,6)(11,12)(13,14), (2,10)(3,9), (2,10,3,4,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
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 \(460800\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6\times F_5$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $S_3\times F_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_2^2\times D_6\times F_5$ | ||
| Normal closure: | $A_5^2:C_2^3$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $D_6\times S_5$ | $C_2\times D_6\times F_5$ | $C_2\times D_6\times F_5$ |
Other information
| Number of subgroups in this autjugacy class | $480$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $S_5^2:D_4$ |