Subgroup ($H$) information
| Description: | $C_5^7:(C_3\times S_3)$ |
| Order: | \(1406250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{7} \) |
| Index: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,45,9,4,43,7,2,41,10,5,44,8,3,42,6)(16,20,19,18,17)(21,23,25,22,24)(26,38,31,27,39,32,28,40,33,29,36,34,30,37,35) \!\cdots\! \rangle$
|
| 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: | $C_5^8.\He_3.C_6$ |
| Order: | \(63281250\)\(\medspace = 2 \cdot 3^{4} \cdot 5^{8} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(4050000000\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | $C_5^6.C_6^3.C_2^3.C_2^4$ |
| $\card{W}$ | 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 | $15$ |
| Möbius function | not computed |
| Projective image | not computed |