Subgroup ($H$) information
| Description: | $C_4\times D_7$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a, d^{2}, c^{4}, d$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $(Q_8\times D_{14}):C_2$ |
| Order: | \(448\)\(\medspace = 2^{6} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{14}.(C_6\times D_4).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2\times D_{14}$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_4\times D_{14}$ |