Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$b^{28}, b^{8}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{14}^2.D_4$ |
| 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: | $C_2^2:C_{28}$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $C_2^5:C_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times C_7:C_3).C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{14}\times C_{56}$ | |||||
| Normalizer: | $C_{14}^2.D_4$ | |||||
| Minimal over-subgroups: | $C_7\times C_{14}$ | $C_2\times C_{14}$ | $C_{28}$ | $C_{28}$ | $C_7:C_4$ | $C_7:C_4$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{14}:C_{28}$ |