Subgroup ($H$) information
| Description: | $F_9$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,6,7,4)(2,9,5,3), (1,3,2)(4,6,10)(5,9,7), (1,4,9)(2,10,5)(3,6,7), (1,3,6,2,7,9,4,5)(11,12), (1,7)(2,5)(3,9)(4,6)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_6:S_3$ |
| Order: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 30T368.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6:D_6$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $F_9$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $F_9$ | |
| Normal closure: | $A_6:S_3$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $\PGL(2,9)$ | $C_3:F_9$ |
| Maximal under-subgroups: | $C_3^2:C_4$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $30$ |
| Möbius function | $1$ |
| Projective image | $A_6:S_3$ |