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

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

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

Permutation Representation (Magma) :

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

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