Subgroup ($H$) information
| Description: | $C_3^3\times C_6$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(3\) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 9 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
13 & 12 \\
6 & 13
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and elementary for $p = 3$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $S_3\times C_3^4$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2.\PSL(4,3).C_2\times S_3$, of order \(145566720\)\(\medspace = 2^{10} \cdot 3^{7} \cdot 5 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $\card{\operatorname{res}(S)}$ | \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^3\times C_6$ | |
| Normalizer: | $C_3^3\times C_6$ | |
| Normal closure: | $S_3\times C_3^4$ | |
| Core: | $C_3^4$ | |
| Minimal over-subgroups: | $S_3\times C_3^4$ | |
| Maximal under-subgroups: | $C_3^4$ | $C_3^2\times C_6$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $S_3$ |