Subgroup ($H$) information
| Description: | $C_{12}.D_4^2$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 0 & 5 & 0 \\
3 & 5 & 0 & 2 \\
0 & 0 & 5 & 0 \\
0 & 0 & 3 & 2
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 0 \\
0 & 6 & 0 & 0 \\
0 & 0 & 6 & 0 \\
0 & 0 & 0 & 6
\end{array}\right), \left(\begin{array}{rrrr}
5 & 5 & 4 & 2 \\
3 & 2 & 3 & 4 \\
1 & 0 & 4 & 2 \\
2 & 1 & 4 & 1
\end{array}\right), \left(\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{array}\right), \left(\begin{array}{rrrr}
0 & 4 & 6 & 3 \\
6 & 1 & 6 & 6 \\
2 & 3 & 6 & 3 \\
4 & 2 & 1 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 5 & 4 & 2 \\
3 & 6 & 3 & 4 \\
1 & 0 & 1 & 2 \\
2 & 1 & 4 & 5
\end{array}\right), \left(\begin{array}{rrrr}
0 & 0 & 4 & 1 \\
6 & 2 & 1 & 6 \\
4 & 6 & 6 & 1 \\
1 & 1 & 3 & 6
\end{array}\right), \left(\begin{array}{rrrr}
0 & 0 & 0 & 2 \\
0 & 4 & 3 & 0 \\
0 & 5 & 3 & 0 \\
5 & 0 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 4 & 6 & 3 \\
1 & 0 & 1 & 6 \\
5 & 0 & 3 & 3 \\
3 & 5 & 6 & 2
\end{array}\right)$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $\GL(2,7):D_8$ |
| Order: | \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_4).C_2^5.\SO(3,7)$ |
| $\operatorname{Aut}(H)$ | $C_4^3.C_2^6$ |
| $W$ | $D_8^2$, of order \(256\)\(\medspace = 2^{8} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $21$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |