Subgroup ($H$) information
| Description: | $C_3\times C_9$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(3\) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$a, b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3\times C_{27}$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(27\)\(\medspace = 3^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_{18}):S_3$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3\times C_{27}$ | |||
| Normalizer: | $C_3\times C_{27}$ | |||
| Minimal over-subgroups: | $C_3\times C_{27}$ | |||
| Maximal under-subgroups: | $C_3^2$ | $C_9$ | $C_9$ | $C_9$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_3$ |