Subgroup ($H$) information
| Description: | $C_5:F_5$ |
| Order: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(1,4,5,2,3), (6,8,7,9,10), (2,3,5,4)(7,9)(8,10), (1,2)(4,5)\rangle$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, supersolvable (hence 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.
Quotient group ($Q$) structure
| Description: | $A_5$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $0$ |
The quotient is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
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 \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $F_5^2$, of order \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| $W$ | $C_5:F_5$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $A_5$ | ||
| Normalizer: | $A_5\times C_5:F_5$ | ||
| Complements: | $A_5$ | ||
| Minimal over-subgroups: | $C_5^2:C_{20}$ | $C_{15}:F_5$ | $C_{10}:F_5$ |
| Maximal under-subgroups: | $C_5\times D_5$ | $C_5:C_4$ | $F_5$ |
Other information
| Möbius function | $-60$ |
| Projective image | $A_5\times C_5:F_5$ |