Subgroup ($H$) information
| Description: | $C_7^2:(C_2^2\times F_7)$ |
| Order: | \(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 6 & 0 & 0 \\
2 & 0 & 6 & 0 \\
0 & 5 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 1 \\
0 & 6 & 3 & 0 \\
0 & 1 & 3 & 0 \\
3 & 0 & 0 & 3
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
6 & 1 & 1 & 0 \\
2 & 0 & 6 & 0 \\
0 & 2 & 1 & 6
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 0 \\
0 & 6 & 0 & 0 \\
0 & 0 & 6 & 0 \\
0 & 0 & 0 & 6
\end{array}\right), \left(\begin{array}{rrrr}
3 & 4 & 6 & 0 \\
5 & 1 & 4 & 6 \\
3 & 1 & 1 & 3 \\
1 & 3 & 2 & 6
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 2 & 6 & 0 \\
4 & 0 & 4 & 0 \\
4 & 1 & 5 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 4 & 1 & 3 \\
5 & 0 & 5 & 1 \\
1 & 4 & 2 & 3 \\
2 & 1 & 2 & 0
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_{42}.D_7^2:S_3$ |
| Order: | \(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times \He_7.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $\He_7.C_6^2.C_2^3$ |
| $W$ | $C_7^2:D_{14}:S_3$, of order \(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $\He_7:C_6\wr C_2$ |