Subgroup ($H$) information
| Description: | $C_{15}\times C_{30}$ |
| Order: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$b^{30}, b^{40}, a^{3}, a^{10}, b^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), maximal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{15}\times C_{60}$ |
| Order: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \GL(2,3)\times \GL(2,5)$ |
| $\operatorname{Aut}(H)$ | $\GL(2,3)\times \GL(2,5)$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $\GL(2,3)\times \GL(2,5)$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{15}\times C_{60}$ | ||
| Normalizer: | $C_{15}\times C_{60}$ | ||
| Minimal over-subgroups: | $C_{15}\times C_{60}$ | ||
| Maximal under-subgroups: | $C_{15}^2$ | $C_5\times C_{30}$ | $C_3\times C_{30}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2$ |