Subgroup ($H$) information
| Description: | $(C_2\times C_4^2):S_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(8505\)\(\medspace = 3^{5} \cdot 5 \cdot 7 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrrrr}
1 & 2 & 0 & 1 & 1 & 0 \\
2 & 0 & 0 & 0 & 0 & 0 \\
0 & 2 & 1 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 \\
0 & 2 & 2 & 1 & 1 & 2 \\
0 & 2 & 2 & 1 & 1 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 0 & 0 & 0 & 0 & 0 \\
1 & 2 & 0 & 1 & 1 & 0 \\
0 & 2 & 2 & 1 & 2 & 0 \\
0 & 2 & 0 & 0 & 2 & 0 \\
2 & 1 & 0 & 1 & 2 & 0 \\
2 & 2 & 0 & 0 & 1 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 2 & 1 & 1 & 2 & 0 \\
2 & 1 & 1 & 0 & 0 & 2 \\
2 & 0 & 0 & 1 & 2 & 2 \\
2 & 2 & 0 & 2 & 2 & 1 \\
1 & 2 & 2 & 2 & 1 & 2 \\
0 & 1 & 2 & 0 & 1 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 0 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 & 0 \\
0 & 2 & 1 & 0 & 0 & 0 \\
1 & 2 & 0 & 1 & 0 & 0 \\
2 & 1 & 0 & 1 & 2 & 0 \\
2 & 2 & 1 & 1 & 0 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 0 & 2 & 0 & 1 & 2 \\
1 & 1 & 1 & 2 & 1 & 1 \\
2 & 1 & 1 & 1 & 2 & 2 \\
2 & 1 & 1 & 0 & 0 & 1 \\
1 & 2 & 0 & 2 & 1 & 0 \\
2 & 1 & 0 & 1 & 1 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 2 & 1 & 1 & 1 & 1 \\
2 & 1 & 0 & 2 & 2 & 0 \\
2 & 0 & 0 & 0 & 1 & 1 \\
2 & 1 & 0 & 0 & 2 & 1 \\
0 & 2 & 0 & 1 & 1 & 2 \\
1 & 1 & 1 & 0 & 2 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
0 & 1 & 2 & 1 & 2 & 2 \\
2 & 2 & 2 & 1 & 2 & 2 \\
2 & 2 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 1 & 0 & 0 \\
1 & 2 & 1 & 2 & 2 & 1 \\
1 & 1 & 1 & 2 & 2 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 1 & 1 & 2 & 0 & 1 \\
1 & 1 & 1 & 2 & 1 & 1 \\
0 & 0 & 2 & 0 & 2 & 0 \\
0 & 1 & 1 & 1 & 2 & 1 \\
2 & 0 & 2 & 2 & 2 & 2 \\
2 & 1 & 0 & 1 & 1 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 0 & 2 & 0 & 2 & 2 \\
1 & 2 & 2 & 1 & 0 & 2 \\
2 & 0 & 1 & 2 & 1 & 2 \\
2 & 0 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 2 & 0 \\
1 & 0 & 0 & 1 & 1 & 2
\end{array}\right) \right]$
|
| Derived length: | $4$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $\PSOMinus(6,3)$ |
| Order: | \(6531840\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5 \cdot 7 \) |
| Exponent: | \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGammaU(4,3)$, of order \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_4^2.(D_4\times S_4).C_2$ |
| $W$ | $(C_2\times C_4^2):S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $8505$ |
| Möbius function | not computed |
| Projective image | $\PSOMinus(6,3)$ |