Subgroup ($H$) information
| Description: | $C_4\times D_7$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a, b^{28}, b^{8}, b^{14}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{28}.D_{14}$ |
| Order: | \(784\)\(\medspace = 2^{4} \cdot 7^{2} \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $D_7$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^2.C_6^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_7:C_8$ | |||
| Normalizer: | $C_{28}.D_{14}$ | |||
| Minimal over-subgroups: | $D_7\times C_{28}$ | $C_8\times D_7$ | ||
| Maximal under-subgroups: | $D_{14}$ | $C_{28}$ | $C_7:C_4$ | $C_2\times C_4$ |
Other information
| Möbius function | $7$ |
| Projective image | $D_7^2$ |