Subgroup ($H$) information
| Description: | $C_{45}$ |
| Order: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Generators: |
$b^{25}, b^{30}, b^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, cyclic (hence abelian, elementary ($p = 3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $D_{45}$ |
| Order: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{45}:C_{12}$, of order \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{45}$ | |
| Normalizer: | $D_{45}$ | |
| Complements: | $C_2$ | |
| Minimal over-subgroups: | $D_{45}$ | |
| Maximal under-subgroups: | $C_{15}$ | $C_9$ |
Other information
| Möbius function | $-1$ |
| Projective image | $D_{45}$ |