Subgroup ($H$) information
| Description: | $F_9$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(6,8,12,14)(9,10,11,15), (6,9,8,10,12,11,14,15), (6,11,14)(8,9,12)(10,15,13), (6,12)(8,14)(9,11)(10,15), (6,13,12)(8,11,10)(9,14,15)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_5$ | ||
| Normalizer: | $F_9:C_{10}$ | ||
| Normal closure: | $\PGL(2,9)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\PGL(2,9)$ | $C_5\times F_9$ | $F_9:C_2$ |
| Maximal under-subgroups: | $C_3^2:C_4$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $-1$ |
| Projective image | $S_6:C_{10}$ |