Subgroup ($H$) information
| Description: | $C_7\times C_{14}$ |
| Order: | \(98\)\(\medspace = 2 \cdot 7^{2} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$b^{14}c^{21}, c^{4}, b^{4}c^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 7$ (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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7.(C_6^2\times D_4).C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(2,7)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_6^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_4.D_{28}$ |