Subgroup ($H$) information
| Description: | $\SL(2,7):C_{12}$ |
| Order: | \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{array}\right), \left(\begin{array}{rrrr}
5 & 2 & 3 & 5 \\
4 & 1 & 4 & 3 \\
6 & 0 & 6 & 5 \\
5 & 6 & 3 & 2
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 0 \\
0 & 6 & 0 & 0 \\
0 & 0 & 6 & 0 \\
0 & 0 & 0 & 6
\end{array}\right), \left(\begin{array}{rrrr}
5 & 4 & 6 & 3 \\
1 & 4 & 1 & 6 \\
5 & 0 & 0 & 3 \\
3 & 5 & 6 & 6
\end{array}\right), \left(\begin{array}{rrrr}
5 & 2 & 4 & 5 \\
3 & 3 & 2 & 6 \\
0 & 2 & 4 & 0 \\
6 & 4 & 1 & 2
\end{array}\right), \left(\begin{array}{rrrr}
1 & 1 & 1 & 4 \\
5 & 1 & 6 & 1 \\
0 & 0 & 5 & 2 \\
4 & 4 & 0 & 0
\end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3\times \SL(2,7).D_8$ |
| Order: | \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_4).C_2^4.\SO(3,7)$ |
| $\operatorname{Aut}(H)$ | $C_2^3\times \PGL(2,7)$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $W$ | $C_2\times \GL(3,2)$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | $D_8\times \GL(3,2)$ |