Polytope of Type {3,2,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6}*72
if this polytope has a name.
Group : SmallGroup(72,46)
Rank : 4
Schlafli Type : {3,2,6}
Number of vertices, edges, etc : 3, 3, 6, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,6,2} of size 144
   {3,2,6,3} of size 216
   {3,2,6,4} of size 288
   {3,2,6,3} of size 288
   {3,2,6,4} of size 288
   {3,2,6,4} of size 288
   {3,2,6,4} of size 432
   {3,2,6,6} of size 432
   {3,2,6,6} of size 432
   {3,2,6,6} of size 432
   {3,2,6,8} of size 576
   {3,2,6,4} of size 576
   {3,2,6,6} of size 576
   {3,2,6,9} of size 648
   {3,2,6,3} of size 648
   {3,2,6,6} of size 648
   {3,2,6,4} of size 720
   {3,2,6,5} of size 720
   {3,2,6,6} of size 720
   {3,2,6,5} of size 720
   {3,2,6,5} of size 720
   {3,2,6,10} of size 720
   {3,2,6,12} of size 864
   {3,2,6,12} of size 864
   {3,2,6,12} of size 864
   {3,2,6,3} of size 864
   {3,2,6,12} of size 864
   {3,2,6,4} of size 864
   {3,2,6,14} of size 1008
   {3,2,6,15} of size 1080
   {3,2,6,16} of size 1152
   {3,2,6,4} of size 1152
   {3,2,6,6} of size 1152
   {3,2,6,3} of size 1152
   {3,2,6,8} of size 1152
   {3,2,6,4} of size 1152
   {3,2,6,12} of size 1152
   {3,2,6,8} of size 1152
   {3,2,6,12} of size 1152
   {3,2,6,6} of size 1152
   {3,2,6,8} of size 1152
   {3,2,6,4} of size 1296
   {3,2,6,12} of size 1296
   {3,2,6,12} of size 1296
   {3,2,6,18} of size 1296
   {3,2,6,18} of size 1296
   {3,2,6,6} of size 1296
   {3,2,6,6} of size 1296
   {3,2,6,6} of size 1296
   {3,2,6,12} of size 1296
   {3,2,6,6} of size 1296
   {3,2,6,20} of size 1440
   {3,2,6,4} of size 1440
   {3,2,6,4} of size 1440
   {3,2,6,4} of size 1440
   {3,2,6,5} of size 1440
   {3,2,6,6} of size 1440
   {3,2,6,6} of size 1440
   {3,2,6,6} of size 1440
   {3,2,6,10} of size 1440
   {3,2,6,10} of size 1440
   {3,2,6,5} of size 1440
   {3,2,6,10} of size 1440
   {3,2,6,10} of size 1440
   {3,2,6,10} of size 1440
   {3,2,6,10} of size 1440
   {3,2,6,15} of size 1440
   {3,2,6,20} of size 1440
   {3,2,6,21} of size 1512
   {3,2,6,22} of size 1584
   {3,2,6,24} of size 1728
   {3,2,6,24} of size 1728
   {3,2,6,24} of size 1728
   {3,2,6,8} of size 1728
   {3,2,6,6} of size 1728
   {3,2,6,6} of size 1728
   {3,2,6,12} of size 1728
   {3,2,6,12} of size 1728
   {3,2,6,3} of size 1800
   {3,2,6,10} of size 1800
   {3,2,6,26} of size 1872
   {3,2,6,9} of size 1944
   {3,2,6,18} of size 1944
   {3,2,6,27} of size 1944
   {3,2,6,6} of size 1944
   {3,2,6,6} of size 1944
   {3,2,6,9} of size 1944
   {3,2,6,9} of size 1944
   {3,2,6,9} of size 1944
   {3,2,6,18} of size 1944
   {3,2,6,3} of size 1944
   {3,2,6,18} of size 1944
Vertex Figure Of :
   {2,3,2,6} of size 144
   {3,3,2,6} of size 288
   {4,3,2,6} of size 288
   {6,3,2,6} of size 432
   {4,3,2,6} of size 576
   {6,3,2,6} of size 576
   {5,3,2,6} of size 720
   {8,3,2,6} of size 1152
   {12,3,2,6} of size 1152
   {6,3,2,6} of size 1296
   {5,3,2,6} of size 1440
   {10,3,2,6} of size 1440
   {10,3,2,6} of size 1440
   {6,3,2,6} of size 1728
   {12,3,2,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,3}*36
   3-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,12}*144, {6,2,6}*144
   3-fold covers : {3,2,18}*216, {9,2,6}*216, {3,6,6}*216a, {3,6,6}*216b
   4-fold covers : {3,2,24}*288, {6,2,12}*288, {12,2,6}*288, {6,4,6}*288, {3,4,6}*288
   5-fold covers : {3,2,30}*360, {15,2,6}*360
   6-fold covers : {3,2,36}*432, {9,2,12}*432, {3,6,12}*432a, {6,2,18}*432, {18,2,6}*432, {6,6,6}*432a, {3,6,12}*432b, {6,6,6}*432b, {6,6,6}*432c, {6,6,6}*432g
   7-fold covers : {3,2,42}*504, {21,2,6}*504
   8-fold covers : {3,2,48}*576, {12,2,12}*576, {6,4,12}*576, {12,4,6}*576, {6,2,24}*576, {24,2,6}*576, {6,8,6}*576, {3,4,12}*576, {3,8,6}*576, {6,4,6}*576a, {6,4,6}*576b
   9-fold covers : {9,2,18}*648, {3,6,18}*648a, {9,6,6}*648a, {3,2,54}*648, {27,2,6}*648, {3,6,6}*648a, {3,6,6}*648b, {3,6,18}*648b, {9,6,6}*648b, {3,6,6}*648c, {3,6,6}*648d, {3,6,6}*648e
   10-fold covers : {15,2,12}*720, {3,2,60}*720, {6,10,6}*720, {6,2,30}*720, {30,2,6}*720
   11-fold covers : {3,2,66}*792, {33,2,6}*792
   12-fold covers : {3,2,72}*864, {9,2,24}*864, {3,6,24}*864a, {6,2,36}*864, {36,2,6}*864, {12,2,18}*864, {18,2,12}*864, {6,6,12}*864a, {12,6,6}*864a, {6,4,18}*864, {18,4,6}*864, {6,12,6}*864a, {3,6,24}*864b, {3,4,18}*864, {9,4,6}*864, {3,12,6}*864a, {6,6,12}*864b, {6,6,12}*864c, {6,12,6}*864b, {12,6,6}*864b, {12,6,6}*864d, {6,6,12}*864e, {12,6,6}*864e, {6,12,6}*864f, {6,12,6}*864g, {3,6,6}*864, {3,12,6}*864b
   13-fold covers : {3,2,78}*936, {39,2,6}*936
   14-fold covers : {21,2,12}*1008, {3,2,84}*1008, {6,14,6}*1008, {6,2,42}*1008, {42,2,6}*1008
   15-fold covers : {3,2,90}*1080, {45,2,6}*1080, {9,2,30}*1080, {15,2,18}*1080, {3,6,30}*1080a, {15,6,6}*1080a, {3,6,30}*1080b, {15,6,6}*1080b
   16-fold covers : {3,2,96}*1152, {12,4,12}*1152, {6,8,12}*1152a, {12,8,6}*1152a, {6,4,24}*1152a, {24,4,6}*1152a, {6,8,12}*1152b, {12,8,6}*1152b, {6,4,24}*1152b, {24,4,6}*1152b, {6,4,12}*1152a, {12,4,6}*1152a, {12,2,24}*1152, {24,2,12}*1152, {6,16,6}*1152, {6,2,48}*1152, {48,2,6}*1152, {3,8,12}*1152, {3,4,12}*1152, {3,8,6}*1152, {3,4,24}*1152, {6,4,12}*1152b, {12,4,6}*1152b, {6,4,12}*1152c, {12,4,6}*1152c, {6,4,6}*1152a, {6,4,6}*1152b, {6,4,12}*1152d, {12,4,6}*1152d, {6,8,6}*1152a, {6,8,6}*1152b, {6,8,6}*1152c, {6,8,6}*1152d, {3,4,6}*1152b
   17-fold covers : {3,2,102}*1224, {51,2,6}*1224
   18-fold covers : {9,2,36}*1296, {9,6,12}*1296a, {3,6,36}*1296a, {27,2,12}*1296, {3,2,108}*1296, {3,6,12}*1296a, {3,6,12}*1296b, {18,2,18}*1296, {6,6,18}*1296a, {18,6,6}*1296a, {6,2,54}*1296, {54,2,6}*1296, {6,6,6}*1296a, {6,6,6}*1296b, {3,6,36}*1296b, {9,6,12}*1296b, {3,6,12}*1296c, {3,6,12}*1296d, {3,6,12}*1296e, {6,6,18}*1296b, {6,6,18}*1296c, {6,6,18}*1296e, {6,18,6}*1296a, {18,6,6}*1296b, {18,6,6}*1296c, {18,6,6}*1296e, {6,6,6}*1296c, {6,6,6}*1296f, {6,6,6}*1296g, {6,6,6}*1296j, {6,6,6}*1296k, {6,6,6}*1296n, {6,6,6}*1296o, {6,6,6}*1296p, {3,6,12}*1296f, {6,6,6}*1296q, {6,6,6}*1296s
   19-fold covers : {3,2,114}*1368, {57,2,6}*1368
   20-fold covers : {15,2,24}*1440, {3,2,120}*1440, {6,10,12}*1440, {12,10,6}*1440, {6,20,6}*1440, {12,2,30}*1440, {30,2,12}*1440, {6,2,60}*1440, {60,2,6}*1440, {6,4,30}*1440, {30,4,6}*1440, {15,4,6}*1440, {3,4,30}*1440
   21-fold covers : {3,2,126}*1512, {63,2,6}*1512, {9,2,42}*1512, {21,2,18}*1512, {3,6,42}*1512a, {21,6,6}*1512a, {3,6,42}*1512b, {21,6,6}*1512b
   22-fold covers : {33,2,12}*1584, {3,2,132}*1584, {6,22,6}*1584, {6,2,66}*1584, {66,2,6}*1584
   23-fold covers : {3,2,138}*1656, {69,2,6}*1656
   24-fold covers : {3,2,144}*1728, {9,2,48}*1728, {3,6,48}*1728a, {12,2,36}*1728, {36,2,12}*1728, {12,6,12}*1728a, {12,4,18}*1728, {18,4,12}*1728, {6,4,36}*1728, {36,4,6}*1728, {6,12,12}*1728a, {12,12,6}*1728a, {6,2,72}*1728, {72,2,6}*1728, {18,2,24}*1728, {24,2,18}*1728, {6,6,24}*1728a, {24,6,6}*1728a, {6,8,18}*1728, {18,8,6}*1728, {6,24,6}*1728a, {3,6,48}*1728b, {3,4,36}*1728, {3,8,18}*1728, {9,4,12}*1728, {3,12,12}*1728a, {9,8,6}*1728, {3,24,6}*1728a, {6,6,24}*1728b, {6,6,24}*1728c, {6,24,6}*1728b, {24,6,6}*1728b, {24,6,6}*1728d, {6,6,24}*1728e, {24,6,6}*1728e, {12,6,12}*1728b, {12,6,12}*1728e, {12,6,12}*1728f, {6,12,12}*1728b, {6,12,12}*1728c, {12,12,6}*1728b, {12,12,6}*1728f, {6,24,6}*1728f, {6,24,6}*1728g, {6,12,12}*1728g, {12,12,6}*1728g, {6,4,18}*1728a, {18,4,6}*1728a, {6,4,18}*1728b, {18,4,6}*1728b, {6,12,6}*1728a, {6,12,6}*1728b, {3,12,6}*1728, {3,24,6}*1728b, {3,6,12}*1728, {3,12,12}*1728b, {6,6,6}*1728a, {6,6,6}*1728f, {6,6,12}*1728a, {6,12,6}*1728e, {6,12,6}*1728f, {6,12,6}*1728h, {6,12,6}*1728i, {6,12,6}*1728j, {6,12,6}*1728l, {12,6,6}*1728a
   25-fold covers : {3,2,150}*1800, {75,2,6}*1800, {3,10,6}*1800, {15,10,6}*1800, {15,2,30}*1800
   26-fold covers : {39,2,12}*1872, {3,2,156}*1872, {6,26,6}*1872, {6,2,78}*1872, {78,2,6}*1872
   27-fold covers : {9,6,18}*1944a, {3,6,6}*1944a, {9,2,54}*1944, {27,2,18}*1944, {3,6,54}*1944a, {27,6,6}*1944a, {3,6,18}*1944a, {9,6,6}*1944a, {3,6,18}*1944b, {9,6,6}*1944b, {3,2,162}*1944, {81,2,6}*1944, {9,6,18}*1944b, {9,18,6}*1944, {3,6,18}*1944c, {3,6,18}*1944d, {9,6,6}*1944c, {9,6,6}*1944d, {3,6,18}*1944e, {9,6,6}*1944e, {3,6,6}*1944b, {3,6,6}*1944c, {3,6,6}*1944d, {3,6,54}*1944b, {27,6,6}*1944b, {3,6,6}*1944e, {3,6,6}*1944f, {3,6,6}*1944g, {9,6,6}*1944f, {9,6,6}*1944g, {9,6,6}*1944h, {3,6,6}*1944h, {3,18,6}*1944
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := (6,7)(8,9);;
s3 := (4,8)(5,6)(7,9);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(9)!(2,3);
s1 := Sym(9)!(1,2);
s2 := Sym(9)!(6,7)(8,9);
s3 := Sym(9)!(4,8)(5,6)(7,9);
poly := sub<Sym(9)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope