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Polytope of Type {3,2,7,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,7,2}*168
if this polytope has a name.
Group : SmallGroup(168,50)
Rank : 5
Schlafli Type : {3,2,7,2}
Number of vertices, edges, etc : 3, 3, 7, 7, 2
Order of s0s1s2s3s4 : 42
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,7,2,2} of size 336
   {3,2,7,2,3} of size 504
   {3,2,7,2,4} of size 672
   {3,2,7,2,5} of size 840
   {3,2,7,2,6} of size 1008
   {3,2,7,2,7} of size 1176
   {3,2,7,2,8} of size 1344
   {3,2,7,2,9} of size 1512
   {3,2,7,2,10} of size 1680
   {3,2,7,2,11} of size 1848
Vertex Figure Of :
   {2,3,2,7,2} of size 336
   {3,3,2,7,2} of size 672
   {4,3,2,7,2} of size 672
   {6,3,2,7,2} of size 1008
   {4,3,2,7,2} of size 1344
   {6,3,2,7,2} of size 1344
   {5,3,2,7,2} of size 1680
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,14,2}*336, {6,2,7,2}*336
   3-fold covers : {9,2,7,2}*504, {3,2,21,2}*504
   4-fold covers : {12,2,7,2}*672, {3,2,28,2}*672, {3,2,14,4}*672, {6,2,14,2}*672
   5-fold covers : {15,2,7,2}*840, {3,2,35,2}*840
   6-fold covers : {9,2,14,2}*1008, {18,2,7,2}*1008, {3,2,14,6}*1008, {3,6,14,2}*1008, {3,2,42,2}*1008, {6,2,21,2}*1008
   7-fold covers : {3,2,49,2}*1176, {3,2,7,14}*1176, {21,2,7,2}*1176
   8-fold covers : {3,2,28,4}*1344, {24,2,7,2}*1344, {3,2,56,2}*1344, {3,2,14,8}*1344, {12,2,14,2}*1344, {6,2,28,2}*1344, {6,2,14,4}*1344, {6,4,14,2}*1344, {3,4,14,2}*1344
   9-fold covers : {27,2,7,2}*1512, {3,2,63,2}*1512, {9,2,21,2}*1512, {3,6,21,2}*1512, {3,2,21,6}*1512
   10-fold covers : {3,2,14,10}*1680, {15,2,14,2}*1680, {30,2,7,2}*1680, {3,2,70,2}*1680, {6,2,35,2}*1680
   11-fold covers : {33,2,7,2}*1848, {3,2,77,2}*1848
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7, 8)( 9,10);;
s3 := (4,5)(6,7)(8,9);;
s4 := (11,12);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(12)!(2,3);
s1 := Sym(12)!(1,2);
s2 := Sym(12)!( 5, 6)( 7, 8)( 9,10);
s3 := Sym(12)!(4,5)(6,7)(8,9);
s4 := Sym(12)!(11,12);
poly := sub<Sym(12)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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