Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$c^{3}, d^{3}$
|
| 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 $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_2\times A_4\times \He_3$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_6\times \He_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_3^4:(S_3\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Outer Automorphisms: | $S_3\times C_3^2:\GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSU(3,2).C_6^2.D_6$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2\times A_4\times \He_3$ |