Subgroup ($H$) information
| Description: | $\He_7:D_8$ |
| Order: | \(5488\)\(\medspace = 2^{4} \cdot 7^{3} \) |
| Index: | \(3\) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 1 & 2 & 0 \\
0 & 1 & 3 & 0 \\
5 & 2 & 6 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 3 & 1 & 0 \\
5 & 4 & 4 & 0 \\
6 & 3 & 0 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 6 & 0 & 0 \\
2 & 0 & 6 & 0 \\
0 & 5 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
5 & 2 & 4 & 5 \\
6 & 4 & 6 & 4 \\
4 & 2 & 5 & 5 \\
1 & 4 & 1 & 4
\end{array}\right), \left(\begin{array}{rrrr}
6 & 0 & 0 & 0 \\
0 & 6 & 0 & 0 \\
3 & 6 & 1 & 0 \\
2 & 3 & 0 & 1
\end{array}\right), \left(\begin{array}{rrrr}
0 & 3 & 0 & 0 \\
2 & 2 & 0 & 0 \\
5 & 6 & 0 & 4 \\
3 & 5 & 5 & 2
\end{array}\right), \left(\begin{array}{rrrr}
1 & 1 & 3 & 3 \\
3 & 4 & 4 & 3 \\
1 & 5 & 5 & 6 \\
5 & 1 & 4 & 1
\end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, and solvable.
Ambient group ($G$) information
| Description: | $\He_7:(C_3\times D_8)$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| 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, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7:(C_3\times \SD_{32})$, of order \(32928\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $\He_7:(C_3\times \SD_{32})$, of order \(32928\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{3} \) |
| $W$ | $\He_7:(C_3\times D_8)$, of order \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $\He_7:(C_3\times D_8)$ | |||
| Complements: | $C_3$ | |||
| Minimal over-subgroups: | $\He_7:(C_3\times D_8)$ | |||
| Maximal under-subgroups: | $\He_7:D_4$ | $\He_7:D_4$ | $\He_7:C_8$ | $D_{56}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $\He_7:(C_3\times D_8)$ |