Subgroup ($H$) information
| Description: | $S_3\times C_{20}$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,4,3,5), (6,8,7,9,10), (2,3)(4,5), (1,11,14)(2,4,3,5)(6,7,10,8,9), (1,14)(6,9,8,10,7)(12,13)\rangle$
|
| 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: | $C_4\times F_5\times S_5$ |
| Order: | \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times D_4\times F_5).S_5$, of order \(38400\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{12}:C_2^3$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $0$ |
| Projective image | $F_5\times S_5$ |