Subgroup ($H$) information
| Description: | $C_{15}^2:D_6$ |
| Order: | \(2700\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(531441\)\(\medspace = 3^{12} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2,3)(7,18,8,16,9,17)(10,38,12,37,11,39)(13,40)(14,41)(15,42)(19,30,21,29,20,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^{12}.C_{15}^2.D_6$ |
| Order: | \(1434890700\)\(\medspace = 2^{2} \cdot 3^{15} \cdot 5^{2} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 \(45916502400\)\(\medspace = 2^{7} \cdot 3^{15} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_{15}^2:D_6$ |
| Normal closure: | $C_3^{12}.C_{15}^2.D_6$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $531441$ |
| Möbius function | not computed |
| Projective image | not computed |