Polytope of Type {3,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,4}*192b
Also Known As : Dual of 1T4(2,0), {{3,4},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(192,1472)
Rank : 4
Schlafli Type : {3,4,4}
Number of vertices, edges, etc : 6, 12, 16, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,4,4,2} of size 384
   {3,4,4,4} of size 768
   {3,4,4,6} of size 1152
   {3,4,4,3} of size 1152
   {3,4,4,6} of size 1728
   {3,4,4,10} of size 1920
Vertex Figure Of :
   {2,3,4,4} of size 384
   {3,3,4,4} of size 768
   {4,3,4,4} of size 768
   {6,3,4,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,4,2}*96
   4-fold quotients : {3,2,4}*48, {3,4,2}*48
   8-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,8,4}*384, {3,4,8}*384, {6,4,4}*384d
   3-fold covers : {9,4,4}*576b, {3,4,12}*576, {3,12,4}*576
   4-fold covers : {3,8,8}*768, {3,4,4}*768a, {3,8,4}*768c, {3,8,4}*768d, {3,4,16}*768, {6,4,4}*768e, {12,4,4}*768e, {12,4,4}*768f, {6,8,4}*768c, {6,4,8}*768c, {6,8,4}*768d
   5-fold covers : {3,4,20}*960, {15,4,4}*960b
   6-fold covers : {9,8,4}*1152, {9,4,8}*1152, {18,4,4}*1152d, {3,8,12}*1152, {3,4,24}*1152, {3,12,8}*1152, {3,24,4}*1152, {6,4,12}*1152c, {6,12,4}*1152i, {6,12,4}*1152j
   7-fold covers : {3,4,28}*1344, {21,4,4}*1344b
   9-fold covers : {27,4,4}*1728b, {3,4,36}*1728, {9,4,12}*1728, {3,12,12}*1728a, {9,12,4}*1728, {3,12,4}*1728a, {3,12,12}*1728b, {3,12,4}*1728b
   10-fold covers : {3,8,20}*1920, {3,4,40}*1920, {15,8,4}*1920, {15,4,8}*1920, {6,4,20}*1920b, {6,20,4}*1920c, {30,4,4}*1920d
Permutation Representation (GAP) : 
s0 := ( 1, 5)( 2, 6)( 7,11)( 8,12);;
s1 := ( 3, 5)( 4, 6)( 9,11)(10,12);;
s2 := ( 3, 4)( 7, 8)(11,12);;
s3 := ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(12)!( 1, 5)( 2, 6)( 7,11)( 8,12);
s1 := Sym(12)!( 3, 5)( 4, 6)( 9,11)(10,12);
s2 := Sym(12)!( 3, 4)( 7, 8)(11,12);
s3 := Sym(12)!( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12);
poly := sub<Sym(12)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3 >;
References :
  1. Theorem 10B3, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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