Polytope of Type {3,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,3}*108
Also Known As : 7T4(1,1)(1,1). if this polytope has another name.
Group : SmallGroup(108,17)
Rank : 4
Schlafli Type : {3,6,3}
Number of vertices, edges, etc : 3, 9, 9, 3
Order of s0s1s2s3 : 3
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,3,2} of size 216
   {3,6,3,4} of size 432
   {3,6,3,6} of size 648
   {3,6,3,4} of size 864
   {3,6,3,8} of size 1728
   {3,6,3,6} of size 1944
Vertex Figure Of :
   {2,3,6,3} of size 216
   {4,3,6,3} of size 432
   {6,3,6,3} of size 648
   {4,3,6,3} of size 864
   {8,3,6,3} of size 1728
   {6,3,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,6}*216a, {6,6,3}*216a
   3-fold covers : {3,6,9}*324, {9,6,3}*324, {3,6,3}*324a, {3,6,3}*324b
   4-fold covers : {3,6,12}*432a, {12,6,3}*432a, {6,6,6}*432a
   5-fold covers : {3,6,15}*540, {15,6,3}*540
   6-fold covers : {3,6,18}*648a, {6,6,9}*648a, {9,6,6}*648a, {18,6,3}*648a, {3,6,6}*648a, {3,6,6}*648b, {6,6,3}*648a, {6,6,3}*648b, {3,6,6}*648e, {6,6,3}*648e
   7-fold covers : {3,6,21}*756, {21,6,3}*756
   8-fold covers : {3,6,24}*864a, {24,6,3}*864a, {6,6,12}*864a, {12,6,6}*864a, {6,12,6}*864a, {3,12,6}*864a, {6,12,3}*864a
   9-fold covers : {9,6,9}*972, {3,6,3}*972, {3,6,27}*972, {27,6,3}*972, {3,6,9}*972a, {9,6,3}*972a, {3,6,9}*972b, {9,6,3}*972b
   10-fold covers : {3,6,30}*1080a, {6,6,15}*1080a, {15,6,6}*1080a, {30,6,3}*1080a
   11-fold covers : {3,6,33}*1188, {33,6,3}*1188
   12-fold covers : {9,6,12}*1296a, {12,6,9}*1296a, {3,6,36}*1296a, {36,6,3}*1296a, {3,6,12}*1296a, {12,6,3}*1296a, {3,6,12}*1296b, {12,6,3}*1296b, {6,6,18}*1296a, {18,6,6}*1296a, {6,6,6}*1296a, {6,6,6}*1296b, {3,6,12}*1296c, {12,6,3}*1296c, {6,6,6}*1296n, {6,6,6}*1296p
   13-fold covers : {3,6,39}*1404, {39,6,3}*1404
   14-fold covers : {3,6,42}*1512a, {6,6,21}*1512a, {21,6,6}*1512a, {42,6,3}*1512a
   15-fold covers : {3,6,45}*1620, {45,6,3}*1620, {9,6,15}*1620, {15,6,9}*1620, {3,6,15}*1620a, {15,6,3}*1620a, {3,6,15}*1620b, {15,6,3}*1620b
   16-fold covers : {3,6,48}*1728a, {48,6,3}*1728a, {12,6,12}*1728a, {6,12,12}*1728a, {12,12,6}*1728a, {6,6,24}*1728a, {24,6,6}*1728a, {6,24,6}*1728a, {3,12,12}*1728a, {12,12,3}*1728a, {3,24,6}*1728a, {6,24,3}*1728a, {3,12,3}*1728, {6,12,6}*1728a, {6,12,6}*1728b
   17-fold covers : {3,6,51}*1836, {51,6,3}*1836
   18-fold covers : {9,6,18}*1944a, {18,6,9}*1944a, {3,6,6}*1944a, {6,6,3}*1944a, {3,6,54}*1944a, {6,6,27}*1944a, {27,6,6}*1944a, {54,6,3}*1944a, {3,6,18}*1944a, {6,6,9}*1944a, {9,6,6}*1944a, {18,6,3}*1944a, {3,6,18}*1944b, {6,6,9}*1944b, {9,6,6}*1944b, {18,6,3}*1944b, {3,6,18}*1944d, {6,6,9}*1944d, {9,6,6}*1944d, {18,6,3}*1944d, {3,6,6}*1944c, {3,6,6}*1944d, {6,6,3}*1944c, {6,6,3}*1944d, {3,6,6}*1944e, {3,6,6}*1944f, {6,6,3}*1944e, {6,6,3}*1944g
Permutation Representation (GAP) : 
s0 := (4,5)(6,7)(8,9);;
s1 := (2,4)(3,6)(8,9);;
s2 := (1,2)(4,9)(5,8)(6,7);;
s3 := (2,3)(4,6)(5,7)(8,9);;
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, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(9)!(4,5)(6,7)(8,9);
s1 := Sym(9)!(2,4)(3,6)(8,9);
s2 := Sym(9)!(1,2)(4,9)(5,8)(6,7);
s3 := Sym(9)!(2,3)(4,6)(5,7)(8,9);
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, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >;
References :
  1. Notes following Theorem 11E7, McMullen P., Schulte, E.; Abstract Regular \ Polytopes (Cambridge University Press, 2002)

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