Subgroup ($H$) information
| Description: | $C_4\times D_7$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a, b^{14}, b^{4}, b^{7}$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, metacyclic (hence supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_4$ | |||
| Normalizer: | $C_4\times D_7$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $D_{14}$ | $C_{28}$ | $C_7:C_4$ | $C_2\times C_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_7$ |