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Polytope of Type {3,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,3,4}*192
Also Known As : hemi-4-cross-polytope, {3,3,4}4if this polytope has another name.
Group : SmallGroup(192,955)
Rank : 4
Schlafli Type : {3,3,4}
Number of vertices, edges, etc : 4, 12, 16, 8
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 4
Special Properties :
   Projective
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,3,4,2} of size 384
   {3,3,4,4} of size 768
   {3,3,4,6} of size 1152
   {3,3,4,10} of size 1920
Vertex Figure Of :
   {2,3,3,4} of size 384
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,3,4}*384, {3,6,4}*384a, {6,3,4}*384a, {3,6,4}*384b, {6,3,4}*384b
   4-fold covers : {3,3,8}*768a, {3,3,8}*768b, {3,12,4}*768a, {12,3,4}*768a, {3,12,4}*768b, {3,6,4}*768a, {3,6,4}*768c, {6,3,4}*768, {6,6,4}*768a, {6,6,4}*768b, {6,6,4}*768c, {3,6,4}*768d, {3,12,4}*768c, {3,12,4}*768d, {12,3,4}*768b, {6,6,4}*768d
   6-fold covers : {3,6,4}*1152a, {3,6,12}*1152, {6,3,4}*1152a
   10-fold covers : {3,6,20}*1920, {6,15,4}*1920, {15,6,4}*1920
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2,10)( 3,11)( 4,12);;
s1 := ( 5, 9)( 6,10)( 7,11)( 8,12);;
s2 := ( 1,11)( 2,12)( 3, 9)( 4,10);;
s3 := ( 3, 4)( 7, 8)(11,12);;
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1*s0*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(12)!( 1, 9)( 2,10)( 3,11)( 4,12);
s1 := Sym(12)!( 5, 9)( 6,10)( 7,11)( 8,12);
s2 := Sym(12)!( 1,11)( 2,12)( 3, 9)( 4,10);
s3 := Sym(12)!( 3, 4)( 7, 8)(11,12);
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1*s0*s2 >; 
 
References : None.
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