Subgroup ($H$) information
| Description: | $C_5\times A_5$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(12,14,15), (6,8,7,9,10), (6,9,8,10,7)(11,12)(13,14)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, an A-group, and nonsolvable.
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.
Quotient group ($Q$) structure
| Description: | $F_5$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times F_5^2$, of order \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_4\times S_5$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $C_2\times A_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $A_5\times C_5:F_5$ |