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Polytope of Type {7,2,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {7,2,3}*84
if this polytope has a name.
Group : SmallGroup(84,8)
Rank : 4
Schlafli Type : {7,2,3}
Number of vertices, edges, etc : 7, 7, 3, 3
Order of s0s1s2s3 : 21
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {7,2,3,2} of size 168
   {7,2,3,3} of size 336
   {7,2,3,4} of size 336
   {7,2,3,6} of size 504
   {7,2,3,4} of size 672
   {7,2,3,6} of size 672
   {7,2,3,5} of size 840
   {7,2,3,8} of size 1344
   {7,2,3,12} of size 1344
   {7,2,3,6} of size 1512
   {7,2,3,5} of size 1680
   {7,2,3,10} of size 1680
   {7,2,3,10} of size 1680
Vertex Figure Of :
   {2,7,2,3} of size 168
   {14,7,2,3} of size 1176
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {7,2,6}*168, {14,2,3}*168
   3-fold covers : {7,2,9}*252, {21,2,3}*252
   4-fold covers : {7,2,12}*336, {28,2,3}*336, {14,2,6}*336
   5-fold covers : {7,2,15}*420, {35,2,3}*420
   6-fold covers : {7,2,18}*504, {14,2,9}*504, {14,6,3}*504, {21,2,6}*504, {42,2,3}*504
   7-fold covers : {49,2,3}*588, {7,2,21}*588
   8-fold covers : {7,2,24}*672, {56,2,3}*672, {14,2,12}*672, {28,2,6}*672, {14,4,6}*672, {14,4,3}*672
   9-fold covers : {7,2,27}*756, {63,2,3}*756, {21,2,9}*756, {21,6,3}*756
   10-fold covers : {7,2,30}*840, {14,2,15}*840, {35,2,6}*840, {70,2,3}*840
   11-fold covers : {7,2,33}*924, {77,2,3}*924
   12-fold covers : {7,2,36}*1008, {28,2,9}*1008, {14,2,18}*1008, {28,6,3}*1008, {21,2,12}*1008, {84,2,3}*1008, {14,6,6}*1008a, {14,6,6}*1008b, {42,2,6}*1008
   13-fold covers : {7,2,39}*1092, {91,2,3}*1092
   14-fold covers : {49,2,6}*1176, {98,2,3}*1176, {7,14,6}*1176, {7,2,42}*1176, {14,2,21}*1176
   15-fold covers : {7,2,45}*1260, {35,2,9}*1260, {21,2,15}*1260, {105,2,3}*1260
   16-fold covers : {7,2,48}*1344, {112,2,3}*1344, {28,2,12}*1344, {14,4,12}*1344, {28,4,6}*1344, {14,2,24}*1344, {56,2,6}*1344, {14,8,6}*1344, {28,4,3}*1344, {14,8,3}*1344, {14,4,6}*1344
   17-fold covers : {7,2,51}*1428, {119,2,3}*1428
   18-fold covers : {7,2,54}*1512, {14,2,27}*1512, {14,6,9}*1512, {14,6,3}*1512, {63,2,6}*1512, {126,2,3}*1512, {21,2,18}*1512, {42,2,9}*1512, {21,6,6}*1512a, {42,6,3}*1512a, {21,6,6}*1512b, {42,6,3}*1512b
   19-fold covers : {7,2,57}*1596, {133,2,3}*1596
   20-fold covers : {7,2,60}*1680, {28,2,15}*1680, {35,2,12}*1680, {140,2,3}*1680, {14,10,6}*1680, {14,2,30}*1680, {70,2,6}*1680
   21-fold covers : {49,2,9}*1764, {147,2,3}*1764, {7,2,63}*1764, {21,2,21}*1764
   22-fold covers : {7,2,66}*1848, {14,2,33}*1848, {77,2,6}*1848, {154,2,3}*1848
   23-fold covers : {7,2,69}*1932, {161,2,3}*1932
Permutation Representation (GAP) :
s0 := (2,3)(4,5)(6,7);;
s1 := (1,2)(3,4)(5,6);;
s2 := ( 9,10);;
s3 := (8,9);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(10)!(2,3)(4,5)(6,7);
s1 := Sym(10)!(1,2)(3,4)(5,6);
s2 := Sym(10)!( 9,10);
s3 := Sym(10)!(8,9);
poly := sub<Sym(10)|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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 

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