Subgroup ($H$) information
| Description: | $\He_3:C_3^2$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$a^{2}, bc^{2}e^{5}, d$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3^3\times C_9):C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^4.D_6$, of order \(708588\)\(\medspace = 2^{2} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.S_3^3$, of order \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(81\)\(\medspace = 3^{4} \) |
| $W$ | $C_3^3:C_6$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $(C_3^3\times C_9):C_6$ |