Subgroup ($H$) information
| Description: | $C_7:C_{21}$ |
| Order: | \(147\)\(\medspace = 3 \cdot 7^{2} \) |
| Index: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 0 & 6 & 0 \\
0 & 2 & 4 & 6 \\
1 & 0 & 0 & 0 \\
4 & 1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
6 & 6 & 5 & 1 \\
4 & 3 & 4 & 5 \\
5 & 6 & 6 & 1 \\
3 & 5 & 3 & 3
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
3 & 1 & 2 & 0 \\
5 & 0 & 4 & 0 \\
6 & 6 & 6 & 2
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $\He_7:(C_3^2\times D_6)$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.(C_6\times S_3^2).C_2$ |
| $\operatorname{Aut}(H)$ | $C_6\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_3\times F_7$, of order \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $28$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\He_7:(C_3^2\times D_6)$ |