Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$c^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{56}.D_{14}$ |
| 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: | $D_7\times D_{14}$ |
| Order: | \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $F_7^2:D_4$, of order \(14112\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{2} \) |
| Outer Automorphisms: | $C_6\wr C_2$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^2.C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(28224\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{56}:C_{14}$ | |||||||||||
| Normalizer: | $C_{56}.D_{14}$ | |||||||||||
| Minimal over-subgroups: | $C_{28}$ | $C_{28}$ | $C_{28}$ | $C_{28}$ | $C_{28}$ | $C_8$ | $C_2\times C_4$ | $D_4$ | $Q_8$ | $Q_8$ | $Q_8$ | $C_8$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $-392$ |
| Projective image | $D_7\times D_{28}$ |