n=8;
gram=IdentityMatrix[3n+2]//MatrixForm;
regularcoords=ConstantArray[0,{3,3n+2}]//MatrixForm;
(*Order is X, Y, R
unit circle passes through the centers of circles 1-n
1-n ring
n+1 inner 
n+2 outer
n+3-2n+2 inner dualS
2n+3-3n+2 outer dual*)
For[i=0,i<n,i++,regularcoords[[1,1,i+1]]=Cos[2Pi*i/n];regularcoords[[1,2,i+1]]=Sin[2Pi*i/n];regularcoords[[1,3,i+1]]=Sin[Pi/n]];
regularcoords[[1,3,n+1]]=1-Sin[Pi/n];
regularcoords[[1,3,n+2]]=-1-Sin[Pi/n];
For[i=0,i<n,i++,regularcoords[[1,1,i+n+3]]=(Sec[Pi/n]-Tan[Pi/n])Cos[(2 i+1)Pi/n];regularcoords[[1,2,i+n+3]]=(Sec[Pi/n]-Tan[Pi/n])Sin[(2 i+1)Pi/n];regularcoords[[1,3,i+n+3]]=(Sec[Pi/n]-Tan[Pi/n])Sin[Pi/n]];
For[i=0,i<n,i++,regularcoords[[1,1,i+2n+3]]=(Sec[Pi/n]+Tan[Pi/n])Cos[(2 i+1)Pi/n];regularcoords[[1,2,i+2n+3]]=(Sec[Pi/n]+Tan[Pi/n])Sin[(2 i+1)Pi/n];regularcoords[[1,3,i+2n+3]]=(Sec[Pi/n]+Tan[Pi/n])Sin[Pi/n]];
Graphics[{Table[Circle[{regularcoords[[1,1,i]],regularcoords[[1,2,i]]},Abs[regularcoords[[1,3,i]]]],{i,Length[regularcoords[[1,1]]]}]}]

xyrtoabbc[{x_,y_,r_}]:={r(((x^2)/(r^2))+((y^2)/(r^2))-1),1/r,x/r,y/r}

abbctoxyr[{bt_,b_,h1_,h2_}]:={h1/b,h2/b,1/b}

graphCircles[circs_]:=Graphics[Circle[{#[[1]],#[[2]]},Abs[#[[3]]]]&/@circs]

graphAbbc[circs_]:=graphCircles[abbctoxyr/@circs]

findRoot[tuple_,sigmas_]:=Block[{candidates},
candidates=Select[sigmas,Total[#.tuple]<Total[tuple]&];
If[Length[candidates]==0,tuple,findRoot[candidates[[1]].tuple,sigmas]]]

MatrixForm[mycoords=(xyrtoabbc/@(Transpose[regularcoords[[1]]])//Simplify)[[n+3;;3n+2]]];

W=mycoords//N;

scale[a_]:=({{a, 0, 0, 0}, {0, 1/a, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}})

rotate[θ_]:=({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, Cos[θ], Sin[θ]}, {0, 0, -Sin[θ], Cos[θ]}})

translate[s_,t_]:=({{1, (s^2)+(t^2), 2s, 2t}, {0, 1, 0, 0}, {0, s, 1, 0}, {0, t, 0, 1}})

invert:=({{0, 1, 0, 0}, {1, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}})

invertAboutCircle[x_,y_,r_]:=translate[x,y].scale[r].invert.scale[1/r].translate[-x,-y]

W2=invertAboutCircle[regularcoords[[1]][[1]][[-1]],regularcoords[[1]][[2]][[-1]],regularcoords[[1]][[3]][[-1]]].#&/@W//Chop;

W2//MatrixForm

W2[[All,2]]

sigmas=findGeneratorsFromG[Simplify[prismG[n]]]//Chop

root=findRoot[W2[[All,2]],sigmas]

(faces=FindFace[graphFromG[prismG[n]]])-1