Subgroup ($H$) information
| Description: | $\He_7:C_6$ |
| Order: | \(2058\)\(\medspace = 2 \cdot 3 \cdot 7^{3} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrr}
5 & 0 & 0 \\
3 & 6 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
5 & 6 & 1
\end{array}\right), \left(\begin{array}{rrr}
6 & 0 & 0 \\
2 & 6 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
6 & 0 & 0 \\
0 & 6 & 0 \\
5 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
4 & 0 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_7^2.\GL(2,7)$ |
| Order: | \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| Exponent: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable. Whether it is almost simple 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^2.\GL(2,7)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $\He_7:C_6^2$, of order \(12348\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | not computed |