Subgroup ($H$) information
| Description: | $C_5^2:C_{45}:C_4$ |
| Order: | \(4500\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Generators: |
$ab^{3}c^{2}e^{2}, d^{3}e^{3}, b^{4}c^{5}d^{8}e^{8}, c^{3}, b^{6}d^{6}e^{3}, e^{3}, c^{10}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}\wr S_3:C_4$ |
| Order: | \(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.(C_{12}\times S_3^2)\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_5^3.C_9.C_6.C_2.C_2^3$ |
| $W$ | $C_5^2:C_{45}:C_{12}$, of order \(13500\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{15}\wr S_3:C_4$ |