Subgroup ($H$) information
| Description: | $D_4^2:C_2^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(25515\)\(\medspace = 3^{6} \cdot 5 \cdot 7 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left[ \left(\begin{array}{rrrrrr}
2 & 2 & 2 & 1 & 2 & 2 \\
0 & 1 & 2 & 1 & 2 & 2 \\
2 & 1 & 2 & 0 & 1 & 1 \\
0 & 0 & 0 & 2 & 0 & 0 \\
0 & 2 & 2 & 1 & 1 & 2 \\
1 & 0 & 1 & 2 & 0 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
1 & 2 & 0 & 1 & 1 & 0 \\
2 & 2 & 2 & 2 & 0 & 2 \\
1 & 0 & 0 & 1 & 0 & 1 \\
0 & 2 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0 & 1 & 1 \\
2 & 1 & 0 & 1 & 1 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 0 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 & 0 \\
0 & 2 & 1 & 0 & 0 & 0 \\
1 & 2 & 0 & 1 & 0 & 0 \\
2 & 1 & 0 & 1 & 2 & 0 \\
2 & 2 & 1 & 1 & 0 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 2 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 2 & 2 & 0 & 0 \\
1 & 1 & 2 & 1 & 0 & 0 \\
2 & 1 & 0 & 2 & 1 & 2 \\
1 & 0 & 1 & 2 & 0 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
2 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 \\
2 & 1 & 2 & 0 & 0 & 0 \\
2 & 2 & 0 & 2 & 0 & 0 \\
1 & 1 & 2 & 2 & 2 & 2 \\
2 & 0 & 2 & 1 & 2 & 1
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
0 & 1 & 0 & 2 & 2 & 0 \\
1 & 2 & 1 & 1 & 0 & 1 \\
2 & 0 & 1 & 2 & 0 & 2 \\
0 & 1 & 2 & 0 & 2 & 2 \\
2 & 1 & 2 & 1 & 1 & 2 \\
2 & 2 & 1 & 0 & 2 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
1 & 1 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 & 0 \\
0 & 2 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 1 & 2 & 2 & 1 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array}\right) \right], \left[ \left(\begin{array}{rrrrrr}
1 & 1 & 0 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 & 0 & 0 \\
2 & 2 & 1 & 0 & 0 & 0 \\
2 & 0 & 0 & 1 & 0 & 0 \\
1 & 2 & 2 & 2 & 1 & 2 \\
1 & 1 & 1 & 0 & 2 & 2
\end{array}\right) \right]$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), 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 \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_4.D_4^2:C_2^4$, of order \(4096\)\(\medspace = 2^{12} \) |
| $W$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $25515$ |
| Möbius function | not computed |
| Projective image | $\PSOMinus(6,3)$ |