Subgroup ($H$) information
| Description: | $C_2\times \He_3$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 2 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 2 & 2 & 1 \\
0 & 1 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 2 & 2 & 2 \\
0 & 1 & 0 & 2 \\
0 & 0 & 0 & 1
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_6.C_3^4$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $2$ |
| 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: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $W$ | $C_3^2$, of order \(9\)\(\medspace = 3^{2} \) |
Related subgroups
| Centralizer: | $C_2\times \He_3$ | |
| Normalizer: | $C_6.C_3^4$ | |
| Complements: | $C_3^2$ | |
| Minimal over-subgroups: | $C_6\times \He_3$ | |
| Maximal under-subgroups: | $\He_3$ | $C_3\times C_6$ |
Other information
| Number of subgroups in this autjugacy class | $90$ |
| Number of conjugacy classes in this autjugacy class | $90$ |
| Möbius function | $3$ |
| Projective image | $C_3^4$ |