Subgroup ($H$) information
| Description: | $C_{15}^2:(C_6\times F_5)$ |
| Order: | \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
| Index: | \(15625\)\(\medspace = 5^{6} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2,3,4,5)(6,42,8,44,10,41,7,43,9,45)(11,15,14,13,12)(16,24,19,22,17,25,20,23,18,21) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_5^7.C_{15}^2:(C_2\times C_{12})$ |
| Order: | \(421875000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{9} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial or rational 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 \(81000000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{9} \) |
| $\operatorname{Aut}(H)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_{15}^2:(C_6\times F_5)$ |
| Normal closure: | $C_5^7.C_{15}^2:(C_2\times C_{12})$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $15625$ |
| Möbius function | not computed |
| Projective image | not computed |