Subgroup ($H$) information
| Description: | $C_6\times C_{15}:F_5$ |
| Order: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$ad^{13}e^{10}, d^{6}e^{6}f^{6}, f^{10}, d^{20}e^{10}, b^{9}, f^{3}, b^{6}d^{20}e^{5}f^{5}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^3.(C_4\times S_4)$ |
| Order: | \(324000\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times \GL(2,3)\times C_5^2:C_4.S_5$ |
| $W$ | $D_5:F_5$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $90$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{15}^3.(C_4\times S_4)$ |