Subgroup ($H$) information
| Description: | $C_{15}\times C_{30}$ |
| Order: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(241\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{15}, b^{2410}, a^{6}, a^{20}, b^{723}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, and metacyclic.
Ambient group ($G$) information
| Description: | $C_{3615}:C_{30}$ |
| Order: | \(108450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \cdot 241 \) |
| Exponent: | \(7230\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 241 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{3615}.C_{120}.C_2^3.C_2$ |
| $\operatorname{Aut}(H)$ | $\GL(2,3)\times \GL(2,5)$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $241$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_{241}:C_{30}$ |