Subgroup ($H$) information
| Description: | $C_2^4.(F_5\times S_4)$ |
| Order: | \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,3)(2,5)(4,9)(6,10)(7,12)(8,14)(11,16)(13,15), (18,20,19,21), (4,9)(6,10) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $F_5\times C_2^6:S_4$ |
| Order: | \(30720\)\(\medspace = 2^{11} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^6.C_{15}.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_5^3\times C_{10}).Q_8$, of order \(1105920\)\(\medspace = 2^{13} \cdot 3^{3} \cdot 5 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^4.(F_5\times S_4)$ |
| Normal closure: | $F_5\times C_2^6:S_4$ |
| Core: | $F_5\times C_2^6$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |