Subgroup ($H$) information
| Description: | $C_2\times S_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(6,10)(8,13)(11,12), (8,11)(12,13)(14,15), (6,10)(7,14)(9,15)(11,12), (7,9)(8,12)(11,13), (6,9,14)(7,15,10)(8,11,12)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $S_6:C_{10}$ |
| Order: | \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{10}$ | ||||
| Normalizer: | $C_{10}\times S_4$ | ||||
| Normal closure: | $S_6$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $S_6$ | $C_{10}\times S_4$ | |||
| Maximal under-subgroups: | $C_2\times A_4$ | $S_4$ | $S_4$ | $C_2\times D_4$ | $D_6$ |
Other information
| Number of subgroups in this conjugacy class | $30$ |
| Möbius function | $0$ |
| Projective image | $S_6:C_{10}$ |