Subgroup ($H$) information
| Description: | $C_6\times C_{15}:F_5$ |
| Order: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$b^{3}d^{48}e^{3}f^{6}, d^{12}e^{3}, d^{40}f^{10}, d^{40}f^{5}, c^{3}d^{42}e, e, c^{2}d^{48}e^{4}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_5^3.C_3^3:S_4$ |
| Order: | \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.C_4^2:\He_3.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times \GL(2,3)\times C_5^2:C_4.S_5$ |
| $\card{W}$ | 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 |