Subgroup ($H$) information
| Description: | $D_7$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$ab^{2}, b^{4}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_4\times D_7$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| 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) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $C_4\times D_7$ | |
| Complements: | $C_4$ $C_4$ | |
| Minimal over-subgroups: | $D_{14}$ | |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
| Autjugate subgroups: | 56.4.4.b1.a1 |
Other information
| Möbius function | $0$ |
| Projective image | $C_4\times D_7$ |