Subgroup ($H$) information
| Description: | $C_6^2:D_6$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 2 & 1 & 1 \\
0 & 2 & 0 & 0 \\
2 & 0 & 2 & 0 \\
1 & 1 & 2 & 1
\end{array}\right), \left(\begin{array}{rrrr}
0 & 2 & 0 & 1 \\
0 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 \\
2 & 2 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\
2 & 0 & 2 & 0 \\
1 & 0 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 \\
2 & 0 & 2 & 0 \\
2 & 1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 2 & 0 & 1 \\
0 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 \\
2 & 2 & 2 & 2
\end{array}\right), \left(\begin{array}{rrrr}
0 & 2 & 2 & 1 \\
0 & 1 & 0 & 0 \\
2 & 2 & 0 & 1 \\
1 & 1 & 1 & 0
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and rational.
Ambient group ($G$) information
| Description: | $C_6^2:\GL(2,3)$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times C_3^2:\GL(2,3)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $(C_2^4\times \He_3).D_6.C_2^2$ |
| $\operatorname{res}(S)$ | $C_3^2:D_6\times S_4$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^2:\GL(2,3)$ |