Subgroup ($H$) information
| Description: | $C_7^2:D_{14}$ |
| Order: | \(1372\)\(\medspace = 2^{2} \cdot 7^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 6 & 0 & 0 \\
2 & 0 & 6 & 0 \\
0 & 5 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 1 \\
0 & 6 & 3 & 0 \\
0 & 1 & 3 & 0 \\
3 & 0 & 0 & 3
\end{array}\right), \left(\begin{array}{rrrr}
3 & 4 & 6 & 0 \\
5 & 1 & 4 & 6 \\
3 & 1 & 1 & 3 \\
1 & 3 & 2 & 6
\end{array}\right), \left(\begin{array}{rrrr}
2 & 4 & 1 & 3 \\
5 & 0 & 5 & 1 \\
1 & 4 & 2 & 3 \\
2 & 1 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 0 \\
0 & 6 & 0 & 0 \\
2 & 2 & 1 & 0 \\
0 & 2 & 0 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_7^2:D_{14}:D_6$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.C_6^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\He_7:C_6\wr C_2$, of order \(24696\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{3} \) |
| $W$ | $\He_7:D_4$, of order \(2744\)\(\medspace = 2^{3} \cdot 7^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-2$ |
| Projective image | $C_7^2:D_{14}:D_6$ |