Subgroup ($H$) information
| Description: | $C_{15}:(S_3\times F_5)$ |
| Order: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$a^{3}, b^{2}d^{9}, a^{6}, c^{10}d^{10}, b^{5}, c^{3}d^{3}, d^{10}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^2:(C_6\times F_5)$ |
| Order: | \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\AGL(2,3)\times C_5^2:C_4.S_5$ |
| $W$ | $C_{15}^2:(C_2\times C_{12})$, of order \(5400\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $5$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_{15}^2:(C_6\times F_5)$ |