Subgroup ($H$) information
| Description: | $S_3^3$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
11 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
16 & 15 \\
15 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
23 & 12 \\
6 & 17
\end{array}\right), \left(\begin{array}{rr}
1 & 15 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 18 \\
24 & 13
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $S_3\times D_6^2$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^3:C_2^2.S_4^2$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $S_3\wr S_3$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $S_3\wr S_3$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $S_3^3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $64$ |
| Number of conjugacy classes in this autjugacy class | $64$ |
| Möbius function | $2$ |
| Projective image | $S_3\times D_6^2$ |