Subgroup ($H$) information
| Description: | $C_5^7:D_7$ |
| Order: | \(1093750\)\(\medspace = 2 \cdot 5^{7} \cdot 7 \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,8)(2,9)(3,10)(4,6)(5,7)(11,17)(12,18)(13,19)(14,20)(15,16)(21,45)(22,41) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_5^8.C_2.\SL(2,8)$ |
| Order: | \(393750000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{8} \cdot 7 \) |
| Exponent: | \(630\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(2362500000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{8} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_5^6.C_{217}.C_{30}.C_2^3.C_2$ |
| $\card{W}$ | 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 | $36$ |
| Möbius function | not computed |
| Projective image | not computed |