Subgroup ($H$) information
| Description: | $C_3^2.C_3^3$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(2\) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$a, b, d^{2}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a $3$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3\times C_{18}):C_3^2$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5:S_3^2$, of order \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^5:S_3^2$, of order \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^5:S_3^2$, of order \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3\times \He_3$, of order \(81\)\(\medspace = 3^{4} \) |
Related subgroups
| Centralizer: | $C_6$ | |||
| Normalizer: | $(C_3\times C_{18}):C_3^2$ | |||
| Complements: | $C_2$ | |||
| Minimal over-subgroups: | $(C_3\times C_{18}):C_3^2$ | |||
| Maximal under-subgroups: | $C_3\times \He_3$ | $C_9:C_3^2$ | $C_9:C_3^2$ | $C_3.\He_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_6\times \He_3$ |