Subgroup ($H$) information
| Description: | $C_{15}:F_5$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,4,5,2,3), (6,8,7,9,10), (2,3,5,4)(7,9)(8,10), (1,4)(3,5)(11,15,13), (1,2)(4,5)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_5\times C_5:F_5$ |
| Order: | \(6000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times F_5^2$, of order \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5^2$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| $W$ | $C_{10}:F_5$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $2$ |
| Projective image | $A_5\times C_5:F_5$ |