Subgroup ($H$) information
| Description: | $C_4\times S_3$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,6)(4,9,7)(5,8)(10,14,12,11), (4,7,9), (7,9), (10,12)(11,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $F_5\times S_3\wr C_3$ |
| Order: | \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $135$ |
| Möbius function | $0$ |
| Projective image | $F_5\times S_3\wr C_3$ |