Subgroup ($H$) information
| Description: | $C_2^2\times F_5\times S_4$ |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,7)(14,15), (1,4,2,3,5), (2,3)(4,5), (1,3,4,5,2)(8,9)(10,14)(11,15)(12,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, a direct factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^3\times F_5\times S_4$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times S_4\times C_2^6.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $F_5\times S_4\times C_2^2:S_4$ |
| $\card{\operatorname{res}(S)}$ | \(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $F_5\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $28$ |
| Number of conjugacy classes in this autjugacy class | $28$ |
| Möbius function | not computed |
| Projective image | $C_2\times F_5\times S_4$ |