Subgroup ($H$) information
| Description: | $(C_5\times C_{15}^2):D_6$ |
| Order: | \(13500\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$ad^{25}e^{2}f^{9}, f^{10}, d^{6}f^{3}, d^{20}, f^{3}, ef^{3}, b^{6}, b^{4}d^{16}ef^{11}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $D_5^3.C_3^2:D_6$ |
| Order: | \(108000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $F_5\times C_5^2.\He_3.C_4.C_2^2$ |
| $W$ | $C_{15}^2:(S_3\times F_5)$, of order \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $D_5^3.C_3^2:D_6$ |