Subgroup ($H$) information
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
2 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
2 & 2 & 1 & 0 \\
0 & 0 & 2 & 0 \\
2 & 0 & 1 & 2
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3^4:C_6$ |
| 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: | $\He_3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.(S_3\times C_3^2:\GL(2,3))$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4374\)\(\medspace = 2 \cdot 3^{7} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_3^3\times C_6$ | ||
| Normalizer: | $C_3^4:C_6$ | ||
| Complements: | $\He_3$ | ||
| Minimal over-subgroups: | $C_3^2\times C_6$ | $C_2\times \He_3$ | $C_3^2\times C_6$ |
| Maximal under-subgroups: | $C_3^2$ | $C_6$ | $C_6$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_3\times \He_3$ |