Subgroup ($H$) information
| Description: | $C_9:C_{15}$ |
| Order: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Generators: |
$a, b^{18}, b^{60}, b^{20}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_9:C_{30}$ |
| Order: | \(270\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times C_3^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_4\times C_3^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times C_3^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3^2$, of order \(9\)\(\medspace = 3^{2} \) |
Related subgroups
| Centralizer: | $C_{30}$ | ||||
| Normalizer: | $C_9:C_{30}$ | ||||
| Complements: | $C_2$ | ||||
| Minimal over-subgroups: | $C_9:C_{30}$ | ||||
| Maximal under-subgroups: | $C_3\times C_{15}$ | $C_{45}$ | $C_{45}$ | $C_{45}$ | $C_9:C_3$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_3\times C_6$ |