Subgroup ($H$) information
| Description: | $C_5\times \He_3$ |
| Order: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$a^{2}, bc^{9}d^{3}, c^{10}, d^{10}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_5\times C_{15}^2):C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_3^2:\GL(2,3)$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $W$ | $C_3\times C_6$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $25$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3\times C_5^3:C_6$ |