Subgroup ($H$) information
| Description: | $C_5\times C_{15}$ |
| Order: | \(75\)\(\medspace = 3 \cdot 5^{2} \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$\langle(1,4,5,2,3), (1,5,3,4,2)(11,15,13), (6,8,7,9,10)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 5$ (hence hyperelementary), and metacyclic.
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 \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times \GL(2,5)$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $W$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $0$ |
| Projective image | $A_5\times C_5:F_5$ |