Subgroup ($H$) information
| Description: | $D_4:C_2$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, c^{21}, d^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{168}.C_2^3$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3\times D_{14}$ |
| Order: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Automorphism Group: | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7.(C_{12}\times D_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{84}$ | ||||
| Normalizer: | $C_{168}.C_2^3$ | ||||
| Minimal over-subgroups: | $D_4:C_{14}$ | $D_4:C_6$ | $Q_{16}:C_2$ | $C_4.C_2^3$ | $D_8:C_2$ |
| Maximal under-subgroups: | $C_2\times C_4$ | $D_4$ | $Q_8$ | $D_4$ | $C_2\times C_4$ |
Other information
| Möbius function | $14$ |
| Projective image | $C_{84}:C_2^3$ |