Subgroup ($H$) information
| Description: | $C_7^2:D_{14}$ |
| Order: | \(1372\)\(\medspace = 2^{2} \cdot 7^{3} \) |
| Index: | \(2\) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 1 & 2 & 0 \\
2 & 0 & 6 & 0 \\
6 & 2 & 2 & 6
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 6 & 0 & 0 \\
2 & 0 & 6 & 0 \\
0 & 5 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
4 & 2 & 1 & 0 \\
6 & 3 & 3 & 1 \\
0 & 4 & 6 & 5 \\
5 & 0 & 1 & 5
\end{array}\right), \left(\begin{array}{rrrr}
2 & 4 & 0 & 0 \\
5 & 0 & 0 & 0 \\
2 & 1 & 2 & 3 \\
4 & 2 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
6 & 6 & 5 & 1 \\
4 & 3 & 4 & 5 \\
5 & 6 & 6 & 1 \\
3 & 5 & 3 & 3
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $\He_7:D_4$ |
| Order: | \(2744\)\(\medspace = 2^{3} \cdot 7^{3} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7:(C_3\times D_8)$, of order \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $\He_7:C_6\wr C_2$, of order \(24696\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{res}(S)$ | $\He_7:(C_3\times D_4)$, of order \(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $\He_7:D_4$, of order \(2744\)\(\medspace = 2^{3} \cdot 7^{3} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $\He_7:D_4$ | ||
| Complements: | $C_2$ | ||
| Minimal over-subgroups: | $\He_7:D_4$ | ||
| Maximal under-subgroups: | $C_7^2:D_7$ | $C_7^2:C_{14}$ | $D_7^2$ |
| Autjugate subgroups: | 2744.r.2.a1.a1 |
Other information
| Möbius function | $-1$ |
| Projective image | $\He_7:D_4$ |