Subgroup ($H$) information
| Description: | $C_5^7:D_7$ |
| Order: | \(1093750\)\(\medspace = 2 \cdot 5^{7} \cdot 7 \) |
| Index: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(11,15,14,13,12)(21,24,22,25,23)(36,39,37,40,38), (21,25,24,23,22)(26,27,28,29,30) \!\cdots\! \rangle$
|
| 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: | $C_5^7.(D_5\times \PGL(2,7))$ |
| Order: | \(262500000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{8} \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^8.C_4^2.\SO(3,7)$ |
| $\operatorname{Aut}(H)$ | $C_5^6.C_{217}.C_{30}.C_2^3.C_2$ |
| $\card{W}$ | 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 | $40$ |
| Möbius function | not computed |
| Projective image | not computed |