Subgroup ($H$) information
| Description: | $C_2\times F_5\times A_5$ |
| Order: | \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,11)(6,10,7,9)(12,13), (6,8,7,9,10), (2,3)(4,5), (6,10,7,9), (6,7)(9,10), (1,14,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, an A-group, and nonsolvable.
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.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 \(38400\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5\times S_5$, of order \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| $W$ | $F_5\times S_5$, of order \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_2\times F_5\times S_5$ |