Subgroup ($H$) information
| Description: | $C_{28}.D_4$ |
| Order: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$a, c^{4}, c^{7}, b^{2}, b, c^{14}$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{28}.D_4$ |
| 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_7\times D_4^2$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_7\times D_4^2$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $W$ | $C_{14}:D_4$, of order \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | |||||||
| Normalizer: | $C_{28}.D_4$ | |||||||
| Complements: | $C_1$ | |||||||
| Maximal under-subgroups: | $D_{28}:C_2$ | $Q_8\times C_{14}$ | $C_7:\OD_{16}$ | $Q_8:D_7$ | $Q_8:D_7$ | $C_7:Q_{16}$ | $C_7:Q_{16}$ | $Q_{16}:C_2$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{14}:D_4$ |