Subgroup ($H$) information
| Description: | $C_7^2:Q_{16}$ |
| Order: | \(784\)\(\medspace = 2^{4} \cdot 7^{2} \) |
| Index: | \(2\) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$ac^{21}, c^{8}, c^{42}, c^{28}, b^{7}c, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{56}.D_{14}$ |
| Order: | \(1568\)\(\medspace = 2^{5} \cdot 7^{2} \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^2.C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_7^2.C_6^2.C_2^4$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(28224\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_7\times D_{28}$, of order \(784\)\(\medspace = 2^{4} \cdot 7^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $C_{56}.D_{14}$ | ||||
| Complements: | $C_2$ $C_2$ | ||||
| Minimal over-subgroups: | $C_{56}.D_{14}$ | ||||
| Maximal under-subgroups: | $C_7^2:Q_8$ | $C_7^2:Q_8$ | $C_7:C_{56}$ | $C_7:Q_{16}$ | $C_7:Q_{16}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $D_7\times D_{28}$ |