Subgroup ($H$) information
| Description: | $C_2^4.(F_5\times S_4)$ |
| Order: | \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| Index: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,14,6)(4,8,7)(5,11,13)(9,15,10)(18,21,20,19), (2,4)(6,7)(9,11)(13,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^4.A_8\times F_5$ |
| Order: | \(6451200\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.A_8\times F_5$, of order \(6451200\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $(C_5^3\times C_{10}).Q_8$, of order \(1105920\)\(\medspace = 2^{13} \cdot 3^{3} \cdot 5 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $140$ |
| Möbius function | not computed |
| Projective image | not computed |