Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(2\) |
| Generators: |
$c^{3}, d^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_6^2.C_3^3$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $\He_3:C_3^2$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Automorphism Group: | $C_3^5:C_3.S_3^2$, of order \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| Outer Automorphisms: | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_6^2.C_3^3.D_6$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_9:C_6^2$ | |||
| Normalizer: | $C_6^2.C_3^3$ | |||
| Complements: | $\He_3:C_3^2$ | |||
| Minimal over-subgroups: | $C_2\times C_6$ | $C_2\times C_6$ | $C_2\times C_6$ | $A_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_6^2.C_3^3$ |