Subgroup ($H$) information
| Description: | $C_{56}:C_2$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$a, b^{14}, c^{2}, b^{7}, b^{4}c^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_4.D_{28}$ |
| Order: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_7\times D_4^2$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_4\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2\times D_4\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\times D_{28}$, of order \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $C_4.D_{28}$ | |||
| Complements: | $C_2$ | |||
| Minimal over-subgroups: | $C_4.D_{28}$ | |||
| Maximal under-subgroups: | $D_{28}$ | $C_7:Q_8$ | $C_{56}$ | $\SD_{16}$ |
| Autjugate subgroups: | 224.104.2.d1.a1 |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{28}$ |