Plotting  CCs in 3D for 6-bodies

r[q_]:=Sqrt[q[[1]]^2 + q[[2]]^2 + q[[3]]^2];
d[q1_,q2_]:=r[q1-q2];
pCC[i_, points_] := points[[i+1]]; (* i-0,..,n-1*)
dCC[i_,j_, points_]:=d[points[[i+1]],points[[j+1]]]; (* i-0,..,n-1*)

Six bodies --- non-planar CCs
There are nine classes of non-planar central configurations: CC10 -- CC18. We present their approximate coordinates and pictures below. 

To check the length of segments between points (bodies) qi and qj in CC number noK, someone can write dCC[i, j, noK]

CC10

(* two symmetrical pyramids with a common square base q0q4q2q5;q1 is the summit of the upper pyramid, q3 --- the lower one. Points q1,q3 and (0,0,0) (the center of mass) are collinear *)

no10 = {(* cc10 *)
{-0.0000000003,0.6521576638,0.0000000000}(* q0 *),
{-0.0000000008,0.0000000009,0.6521576638} (* q1 *),
{0.0000000003,-0.6521576638,0.0000000013} (* q2 *),
{0.0000000008,-0.0000000005,-0.6521576636} (* q3 *),
{0.6521576637,0.0000000000,0.0000000000},(* q4 *)
{-0.6521576637,-0.0000000004,-0.0000000014}}; (* q5*)
S10=Polygon[{pCC[0, no10],pCC[4, no10],pCC[2, no10],pCC[5, no10]}];
data = {};
data = Append[data, {Callout[no10[[1]],0]}]; 
data = Append[data, {Callout[no10[[2]],1]}]; 
data = Append[data, {Callout[no10[[3]],2]}]; 
data = Append[data, {Callout[no10[[4]],3]}]; 
data = Append[data, {Callout[no10[[5]],4]}]; 
data = Append[data, {Callout[no10[[6]],5]}]; 

Show[(*ListPlot3D[no10,AspectRatio->Automatic,PlotRange->Full, BoundaryStyle -> Thick, Mesh->All,PlotStyle->Directive[White,Opacity[.1],Specularity[White,40]]],*)ListPointPlot3D[data,AspectRatio->Automatic, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],S10}],
Graphics3D[{Dashed,Line[{pCC[0, no10],pCC[3, no10]}]}],
Graphics3D[Line[{pCC[2, no10],pCC[3, no10]}]],
Graphics3D[Line[{pCC[4, no10],pCC[3, no10]}]],
Graphics3D[Line[{pCC[5, no10],pCC[3, no10]}]],
Graphics3D[{Dashed,Line[{pCC[0, no10],pCC[1, no10]}]}],
Graphics3D[Line[{pCC[2, no10],pCC[1, no10]}]],
Graphics3D[Line[{pCC[4, no10],pCC[1, no10]}]],
Graphics3D[Line[{pCC[5, no10],pCC[1, no10]}]]
](* Show *)

CC11

(* two symmetric tetrahedrons with a common equilateral triangle base q0q1q3;q2=(0,0,0) lies at the triangle plane (i.e.\ bodies q0, q1, q2 and q3 are coplanar);q4 is the summit of right pyramid, q5 --- the left one. Points q2,q4 and q5 are collinear *)

no11 = {(* cc11 *)
{-0.0000000000,0.7240953283,0.0000000000},
{0.0000000000,-0.3620476642,0.6270849491},
{-0.0000000000,-0.0000000000,-0.0000000000},
{0.0000000000,-0.3620476642,-0.6270849491},
{0.7285787808,0.0000000000,0.0000000000},
{-0.7285787808,-0.0000000000,0.0000000000}};
T11=Triangle[{pCC[0,no11], pCC[1,no11], pCC[3, no11]}];
data = {};
data = Append[data, {Callout[no11[[1]],0]}]; 
data = Append[data, {Callout[no11[[2]],1]}]; 
data = Append[data, {Callout[no11[[3]],2]}]; 
data = Append[data, {Callout[no11[[4]],3]}]; 
data = Append[data, {Callout[no11[[5]],4]}]; 
data = Append[data, {Callout[no11[[6]],5]}]; 

Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],T11}],
Graphics3D[Line[{pCC[1, no11],pCC[5, no11]}]],
Graphics3D[Line[{pCC[3, no11],pCC[5, no11]}]],
Graphics3D[Line[{pCC[0, no11],pCC[5, no11]}]],
Graphics3D[Line[{pCC[1, no11],pCC[4, no11]}]],
Graphics3D[Line[{pCC[3, no11],pCC[4, no11]}]],
Graphics3D[Line[{pCC[0, no11],pCC[4, no11]}]],
Graphics3D[{Dashed, Line[{pCC[5, no11],pCC[4, no11]}]}]
](* Show *)

CC12

(* two pyramids (one inside the other) with a common square base q0q1q3q5;q2 is the summit of innner pyramid, q4 --- the outer one. Points q2,q4 and (0,0,0) (the center of mass) are collinear *)

no12 = {(* cc12 *)
{0.7893559716,0.0,0.0} (* c0 *),
{0.0,-0.7893559716,0.0} (* c1 *),
{-0.7893559716,0.0,0.0} (* c2 *),
{0.0,0.7893559716,0.0} (* c3 *),
{0.0,0.0,1.18072},
{0.0,0.0,0.3913642704}
};
S12=Polygon[{pCC[0, no12],pCC[1, no12],pCC[2, no12],pCC[3, no12]}];
data = {};
data = Append[data, {Callout[no12[[1]],0]}]; 
data = Append[data, {Callout[no12[[2]],5]}]; 
data = Append[data, {Callout[no12[[3]],3]}]; 
data = Append[data, {Callout[no12[[4]],1]}]; 
data = Append[data, {Callout[no12[[5]],4]}]; 
data = Append[data, {Callout[no12[[6]],2]}]; 

Show[ListPointPlot3D[data,AspectRatio->Automatic, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],S12}],Graphics3D[Line[{pCC[4, no12],pCC[5, no12]}]],
(* dashed lines *)
Graphics3D[{Dashed,Line[{pCC[5, no12],{0.0, 0.0, 0.0}}]}],Graphics3D[{Dashed,Line[{pCC[1, no12],{0.0, 0.0, 0.0}}]}],Graphics3D[{Dashed,Line[{pCC[2, no12],{0.0, 0.0, 0.0}}]}],Graphics3D[{Dashed,Line[{pCC[3, no12],{0.0, 0.0, 0.0}}]}],Graphics3D[{Dashed,Line[{pCC[0, no12],{0.0, 0.0, 0.0}}]}],
(* outer pyramid *)
Graphics3D[Line[{pCC[0, no12],pCC[4, no12]}]],
Graphics3D[Line[{pCC[1, no12],pCC[4, no12]}]],
Graphics3D[Line[{pCC[3, no12],pCC[4, no12]}]],
Graphics3D[Line[{pCC[2, no12],pCC[4, no12]}]],
(* inner pyramid *)
Graphics3D[Line[{pCC[0, no12],pCC[5, no12]}]],Graphics3D[Line[{pCC[1, no12],pCC[5, no12]}]],Graphics3D[Line[{pCC[3, no12],pCC[5, no12]}]],
Graphics3D[Line[{pCC[2, no12],pCC[5, no12]}]] 
](* Show *)

CC13

(* two orthogonal isoseles triangles; altitudes of both triangles, points q2,q4 and the center of mass (point (0,0,0)) lie on the line marked red on the picture.*)

no13 = {(* cc13 *)
{-0.0836906941,0.7132647832,0.0000000000},
{-0.3752616262,0.0000000000,0.5877313798},
{0.1363832036,-0.0000000000,0.0000000000},
{-0.0836906941,-0.7132647832,0.0000000000},
{0.7815214369,0.0000000000,0.0000000000},
{-0.3752616262,0.0000000000,-0.5877313798}};
T1=Triangle[{pCC[0, no13],pCC[3, no13],pCC[4, no13]}]; T2=Triangle[{pCC[1, no13],pCC[5, no13],pCC[4, no13]}];
 
data = {};
data = Append[data, {Callout[no13[[1]],0]}]; 
data = Append[data, {Callout[no13[[2]],1]}]; 
data = Append[data, {Callout[no13[[3]],2]}]; 
data = Append[data, {Callout[no13[[4]],3]}]; 
data = Append[data, {Callout[no13[[5]],4]}]; 
data = Append[data, {Callout[no13[[6]],5]}]; 

Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full],Graphics3D[{Yellow,Opacity[.1],T1}],
Graphics3D[{Blue,Opacity[.1],T2}],
Graphics3D[{Red,Line[{{-0.3752616262,0.0000000000,0},pCC[4, no13]}]}]
](* Show *)

CC14

(* two pyramids (one inside the other) with a common square base q0q1q3q5;q2 is the summit of innner pyramid,q4 --- the outer one. Points q2,q4 and (0,0,0) (the center of mass) are collinear. *)

(* oryginal data from raport file:*)
no14 = {(* cc14 *)
{-0.1851720170,0.7018558008,0.0000000000},
{-0.1851720170,0.0000000000,0.7018558008},
{0.0138656978,-0.0000000000,0.0000000000},
{-0.1851720170,-0.7018558008,-0.0000000000},
{0.7268223701,0.0000000000,0.0000000000},
{-0.1851720170,0.0000000000,-0.7018558008}};

(* rotated cc14 so that square base is on z=0 plane *)
(*no14 = {(* cc14 *)
{0.7018558008,0.0000000000, 0.0} (* q0 *),
{0.0000000000,0.7018558008, 0.0} (* q1 *),
{0.0000000000,0.0000000000,0.199037714}(* q2 *),
{-0.7018558008,0.0000000000, 0.0}(* q3 *),
{0.0000000000,0.0000000000, 0.9119943870999999}(* q4 *),
{0.0000000000,-0.7018558008, 0.0}(* q5 *)};*)


S14=Polygon[{pCC[0, no14],pCC[1, no14],pCC[3, no14],pCC[5, no14]}];
data = {};
data = Append[data, {Callout[no14[[1]],0]}]; 
data = Append[data, {Callout[no14[[2]],1]}]; 
data = Append[data, {Callout[no14[[3]],2]}]; 
data = Append[data, {Callout[no14[[4]],3]}]; 
data = Append[data, {Callout[no14[[5]],4]}]; 
data = Append[data, {Callout[no14[[6]],5]}]; 

Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full],Graphics3D[{Blue,Opacity[.1],S14}],
(* dashed lines meeting in the center of mass *)
Graphics3D[{Dashed,Line[{pCC[5, no14],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[1, no14],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[4, no14],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[3, no14],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[0, no14],{0.0, 0.0, 0.0}}]}],
(* outer pyramid *)
Graphics3D[Line[{pCC[0, no14],pCC[4, no14]}]],
Graphics3D[Line[{pCC[1, no14],pCC[4, no14]}]],
Graphics3D[Line[{pCC[3, no14],pCC[4, no14]}]],
Graphics3D[Line[{pCC[5, no14],pCC[4, no14]}]],

(* inner pyramid *)
Graphics3D[Line[{pCC[0, no14],pCC[2, no14]}]],
Graphics3D[Line[{pCC[1, no14],pCC[2, no14]}]],
Graphics3D[Line[{pCC[3, no14],pCC[2, no14]}]],
Graphics3D[Line[{pCC[5, no14],pCC[2, no14]}]]
](* Show *)

CC15

(* polyhedron with a regular pentagon q0q4q2q5q3 as a base;q1 is the summit and is collinear with (0,0,0) (the center of mass) *)
no15 = {(* cc15 *) 
{0.2214967272,0.6246039852,0.0000000000},
{-0.1208899775,-0.0853962625,0.6153624889},
{0.2214967272,-0.6010010388,-0.1700820089},
{-0.4924091506,0.4096294388,-0.1700820089},
{0.6627148244,0.0000000000,0.0000000000},
{-0.4924091506,-0.3478361228,-0.2751984712}};;

data = {};
data = Append[data, {Callout[no15[[1]],0]}]; 
data = Append[data, {Callout[no15[[2]],1]}]; 
data = Append[data, {Callout[no15[[3]],2]}]; 
data = Append[data, {Callout[no15[[4]],3]}]; 
data = Append[data, {Callout[no15[[5]],4]}]; 
data = Append[data, {Callout[no15[[6]],5]}]; 
P15=Polygon[{pCC[0, no15],pCC[4, no15],pCC[2, no15],pCC[5, no15],pCC[3, no15]}];


Show[ListPointPlot3D[data,AspectRatio->Full, 
PlotRange->Full] ,Graphics3D[{Blue,Opacity[.1],P15}],
Graphics3D[{Thick, Dashed, Line[{pCC[1, no15],{0.0, 0.0, 0.0}}]}],
(* dashed lines meeting in the center of mass *)
 Graphics3D[{Dashed,Line[{pCC[5, no15],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[2, no15],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[4, no15],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[3, no15],{0.0, 0.0, 0.0}}]}],
Graphics3D[{Dashed,Line[{pCC[0, no15],{0.0, 0.0, 0.0}}]}],
(* pyramid *)
Graphics3D[Line[{pCC[0, no15],pCC[1, no15]}]],
Graphics3D[Line[{pCC[2, no15],pCC[1, no15]}]],
Graphics3D[Line[{pCC[3, no15],pCC[1, no15]}]],
Graphics3D[Line[{pCC[4, no15],pCC[1, no15]}]],
Graphics3D[Line[{pCC[5, no15],pCC[1, no15]}]]
](* Show *)

CC16

(* polyhedron with three rectangular and two triangular faces;q0q1q4 and q2q3q5 are symmetrical equilateral triangles, thus rectangles q0q1q3q5,q0q4q2q5 and q1q3q2q4 are also symmetrical;lines with the same length have the same color (red or blue) *)

no16 = {(* cc16 *) 
{0.0521195982,0.6524780215,0.0000000000},
{0.0521195982,0.0481223285,0.6507010143},
{0.1486926539,-0.4670669000,-0.4338006762},
{-0.4537441023,-0.4189445715,0.2169003381},
{0.6545563544,0.0000000000,0.0000000000},
{-0.4537441023,0.1854111215,-0.4338006762}};

data = {};
data = Append[data, {Callout[no16[[1]],0]}]; 
data = Append[data, {Callout[no16[[2]],1]}]; 
data = Append[data, {Callout[no16[[3]],2]}]; 
data = Append[data, {Callout[no16[[4]],3]}]; 
data = Append[data, {Callout[no16[[5]],4]}]; 
data = Append[data, {Callout[no16[[6]],5]}]; 
R16=Polygon[{pCC[0, no16],pCC[4, no16],pCC[2, no16],pCC[5, no16]}];

Show[ListPointPlot3D[data,AspectRatio->Automatic,
PlotRange->Full] ,Graphics3D[{Blue,Opacity[.1],R16}],

Graphics3D[Line[{pCC[0, no16],pCC[1, no16]}]],
Graphics3D[Line[{pCC[2, no16],pCC[3, no16]}]],
Graphics3D[Line[{pCC[3, no16],pCC[1, no16]}]],
Graphics3D[Line[{pCC[4, no16],pCC[1, no16]}]],
Graphics3D[Line[{pCC[5, no16],pCC[3, no16]}]],
(* lines with the same length have the same color --- blue or red *)
Graphics3D[{Red,Line[{pCC[0, no16],pCC[5, no16]}]}],
Graphics3D[{Red,Line[{pCC[2, no16],pCC[4, no16]}]}],
Graphics3D[{Red,Line[{pCC[1, no16],pCC[3, no16]}]}],
Graphics3D[{ Blue,Line[{pCC[5, no16],pCC[2, no16]}]}],
Graphics3D[{ Blue,Line[{pCC[5, no16],pCC[3, no16]}]}],
Graphics3D[{ Blue,Line[{pCC[3, no16],pCC[2, no16]}]}],
Graphics3D[{ Blue,Line[{pCC[1, no16],pCC[0, no16]}]}],
Graphics3D[{ Blue,Line[{pCC[1, no16],pCC[4, no16]}]}],
Graphics3D[{ Blue,Line[{pCC[0, no16],pCC[4, no16]}]}]
](* Show *)

CC17

(* triangular polyhedron with faces being isoseles triangles;q0q1q4 and q0q1q5 are symmetrical, q0q4q5 has one side longer (the segment (q4,q5) is longer than (q0,q1));triangle q0q2q3 is also isoseles triangle. *)

no17 := {(* cc17 *)
{-0.1657434868,0.6791491244,0.0000000000},(* q0 *)
{-0.1657434868,-0.1394836664,0.6646712270}, (* q1 *)
{0.3203559442,-0.0161277871,-0.0198635578}, (* q2 *)
{-0.2196013590,-0.1479020955,-0.1821614962}, (* q3 *)
{0.8849667893,0.0000000000,0.0000000000}, (* q4 *)
{-0.6542344009,-0.3756355754,-0.4626461730}}; (* q5 *)
smallT17 = Triangle[{no17[[1]], no17[[3]], no17[[4]]}];
bigT17 = Triangle[{no17[[1]], no17[[5]], no17[[6]]}];

data = {};
data = Append[data, {Callout[no17[[1]],0]}]; 
data = Append[data, {Callout[no17[[2]],1]}]; 
data = Append[data, {Callout[no17[[3]],2]}]; 
data = Append[data, {Callout[no17[[4]],3]}]; 
data = Append[data, {Callout[no17[[5]],4]}]; 
data = Append[data, {Callout[no17[[6]],5]}]; 

Show[ListPointPlot3D[data,AspectRatio->Automatic,
PlotRange->Full] , Graphics3D[{Blue,Opacity[.1],bigT17}], Graphics3D[{Green,Opacity[.1],smallT17}],
(* lines with the same length have the same color (red or blue)*)
Graphics3D[{Thickness[0.0045],Blue, Line[{pCC[0, no17],pCC[5, no17]}]}],
Graphics3D[{Thickness[0.0045],Blue, Line[{pCC[1, no17],pCC[4, no17]}]}],
Graphics3D[{Thickness[0.0045],Blue, Line[{pCC[0, no17],pCC[4, no17]}]}],
Graphics3D[Line[{pCC[0, no17],pCC[1, no17]}]],
Graphics3D[Line[{pCC[4, no17],pCC[1, no17]}]],
Graphics3D[{Red,Line[{pCC[3, no17],pCC[0, no17]}]}],
Graphics3D[{Red,Line[{pCC[2, no17],pCC[0, no17]}]}],
Graphics3D[{ Thickness[0.0045],Blue, Line[{pCC[5, no17],pCC[1, no17]}]}]
](* Show *)

CC18

no18 := {(* cc18 *)
{0.0153012472,0.7243074165,0.0000000000},
{-0.2123417131,-0.1865650076,0.6613356770},
{0.0642192369,-0.0045125196,0.0159959805},
{0.0153012472,-0.6175215813,-0.3785344505},
{0.7520348732,0.0000000000,0.0000000000},
{-0.6345148914,0.0842916920,-0.2987972070}};

P18 :=Polygon[{pCC[1, no18],pCC[4, no18],pCC[5, no18]}];
data = {};
data = Append[data, {Callout[no18[[1]],0]}]; 
data = Append[data, {Callout[no18[[2]],1]}]; 
data = Append[data, {Callout[no18[[3]],2]}]; 
data = Append[data, {Callout[no18[[4]],3]}]; 
data = Append[data, {Callout[no18[[5]],4]}]; 
data = Append[data, {Callout[no18[[6]],5]}]; 
data = Append[data, {Callout[{0,0,0},c]}]; 


Show[ListPointPlot3D[data,AspectRatio->Automatic,
PlotRange->Full] , Graphics3D[{Blue,Opacity[.1],P18}],Graphics3D[{ Thickness[0.002],Blue,Line[{pCC[0, no18],pCC[1, no18]}]}],Graphics3D[ {Thickness[0.002],Magenta,Line[{pCC[0, no18],pCC[4, no18]}]}],Graphics3D[ {Thickness[0.002],Red,Line[{pCC[0, no18],pCC[5, no18]}]}],
Graphics3D[{Thickness[0.002],Red,Line[{pCC[3, no18],pCC[5, no18]}]}],
Graphics3D[{Dashed,Thickness[0.002],Blue,Line[{pCC[3, no18],pCC[1, no18]}]}],
Graphics3D[{Thickness[0.002],Magenta,Line[{pCC[4, no18],pCC[3, no18]}]}],
Graphics3D[{Dashed,Thickness[0.002],Green,Line[{pCC[3, no18],pCC[2, no18]}]}],
Graphics3D[{Dashed,Thickness[0.002],Green,Line[{pCC[0, no18],pCC[2, no18]}]}],
Graphics3D[{Dashed,Thickness[0.002],Orange,Line[{pCC[3, no18],{0,0,0}}]}],
Graphics3D[{Dashed,Thickness[0.002],Orange,Line[{pCC[0, no18],{0,0,0}}]}]
](* Show *)