Subgroup ($H$) information
| Description: | $C_{15}:F_5:S_3$ |
| Order: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$ab^{6}c^{3}d^{6}ef^{2}, d^{3}f^{12}, b^{6}c^{8}d^{6}f^{12}, b^{4}d^{6}f^{12}, b^{3}c^{2}d^{6}f^{12}, f^{3}, d^{10}f^{5}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^2.(F_5\times S_3^2)$ |
| Order: | \(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^3.C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $F_{25}:D_6$, of order \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| $W$ | $C_5^2:(C_4\times S_3^2)$, of order \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $45$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_{15}^2.(F_5\times S_3^2)$ |