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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,2,7}*168
if this polytope has a name.
Group : SmallGroup(168,50)
Rank : 5
Schlafli Type : {3,2,2,7}
Number of vertices, edges, etc : 3, 3, 2, 7, 7
Order of s₀s₁s₂s₃s₄ : 42
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,2,7,2} of size 336
Vertex Figure Of :
   {2,3,2,2,7} of size 336
   {3,3,2,2,7} of size 672
   {4,3,2,2,7} of size 672
   {6,3,2,2,7} of size 1008
   {4,3,2,2,7} of size 1344
   {6,3,2,2,7} of size 1344
   {5,3,2,2,7} of size 1680
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,2,14}*336, {6,2,2,7}*336
   3-fold covers : {9,2,2,7}*504, {3,6,2,7}*504, {3,2,2,21}*504
   4-fold covers : {12,2,2,7}*672, {3,2,2,28}*672, {3,2,4,14}*672, {6,4,2,7}*672a, {3,4,2,7}*672, {6,2,2,14}*672
   5-fold covers : {15,2,2,7}*840, {3,2,2,35}*840
   6-fold covers : {9,2,2,14}*1008, {18,2,2,7}*1008, {3,2,6,14}*1008, {3,6,2,14}*1008, {6,6,2,7}*1008a, {6,6,2,7}*1008c, {3,2,2,42}*1008, {6,2,2,21}*1008
   7-fold covers : {3,2,2,49}*1176, {3,2,14,7}*1176, {21,2,2,7}*1176
   8-fold covers : {12,4,2,7}*1344a, {3,2,4,28}*1344, {24,2,2,7}*1344, {3,2,2,56}*1344, {3,2,8,14}*1344, {6,8,2,7}*1344, {3,8,2,7}*1344, {12,2,2,14}*1344, {6,2,2,28}*1344, {6,2,4,14}*1344, {6,4,2,14}*1344a, {3,4,2,14}*1344, {6,4,2,7}*1344
   9-fold covers : {27,2,2,7}*1512, {9,6,2,7}*1512, {3,6,2,7}*1512, {3,2,2,63}*1512, {9,2,2,21}*1512, {3,2,6,21}*1512, {3,6,2,21}*1512
   10-fold covers : {3,2,10,14}*1680, {6,10,2,7}*1680, {15,2,2,14}*1680, {30,2,2,7}*1680, {3,2,2,70}*1680, {6,2,2,35}*1680
   11-fold covers : {33,2,2,7}*1848, {3,2,2,77}*1848
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := (4,5);;
s3 := ( 7, 8)( 9,10)(11,12);;
s4 := ( 6, 7)( 8, 9)(10,11);;
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, 
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(12)!(2,3);
s1 := Sym(12)!(1,2);
s2 := Sym(12)!(4,5);
s3 := Sym(12)!( 7, 8)( 9,10)(11,12);
s4 := Sym(12)!( 6, 7)( 8, 9)(10,11);
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, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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