Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_3^3.C_3^2$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3.\He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Automorphism Group: | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5:D_6$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^3.C_3^2$ | ||
| Normalizer: | $C_3^3.C_3^2$ | ||
| Minimal over-subgroups: | $C_3^2$ | $C_3^2$ | $C_3^2$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_3.\He_3$ |