Subgroup ($H$) information
| Description: | $(C_5\times C_{15}^2):C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$c^{3}d^{27}e^{7}f, e^{2}f^{2}, bd^{9}e^{3}f^{3}, d^{20}, c^{2}, f, d^{6}e^{6}f$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $D_5^3.C_3^2:C_6$ |
| Order: | \(54000\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5^{3} \) |
| 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 \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $W$ | $C_5\wr C_3:C_{12}$, of order \(4500\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $D_5^3.C_3^2:C_6$ |