Subgroup ($H$) information
| Description: | $C_7^2:F_7$ |
| Order: | \(2058\)\(\medspace = 2 \cdot 3 \cdot 7^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 6 & 0 & 0 \\
2 & 0 & 6 & 0 \\
0 & 5 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 4 & 1 & 0 \\
6 & 0 & 2 & 0 \\
3 & 3 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 3 & 5 & 0 \\
2 & 4 & 1 & 5 \\
0 & 6 & 5 & 4 \\
4 & 0 & 5 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 6 & 0 \\
0 & 2 & 4 & 6 \\
1 & 0 & 0 & 0 \\
4 & 1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
6 & 6 & 5 & 1 \\
4 & 3 & 4 & 5 \\
5 & 6 & 6 & 1 \\
3 & 5 & 3 & 3
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
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.
Quotient group ($Q$) structure
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.(C_6\times S_3^2).C_2$ |
| $\operatorname{Aut}(H)$ | $F_7\wr C_2$, of order \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| $W$ | $C_7^2:(C_6\times S_3)$, of order \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $\He_7:(C_3^2\times D_6)$ |