Subgroup ($H$) information
| Description: | $C_2\times C_{14}$ |
| Order: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Index: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$b^{14}c^{21}, b^{4}c^{14}, c^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{28}.D_{28}$ |
| 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: | $C_7\times D_4$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7.(C_6^2\times D_4).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{14}\times C_{28}$ | |||
| Normalizer: | $C_{28}.D_{28}$ | |||
| Minimal over-subgroups: | $C_{14}^2$ | $C_2\times C_{28}$ | $C_7:D_4$ | $C_{14}:C_4$ |
| Maximal under-subgroups: | $C_{14}$ | $C_{14}$ | $C_2^2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{14}\times D_{28}$ |