Polytope of Type {3,2,7}

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

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

to this polytope