Polytope of Type {2,2,21}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,21}*168
if this polytope has a name.
Group : SmallGroup(168,56)
Rank : 4
Schlafli Type : {2,2,21}
Number of vertices, edges, etc : 2, 2, 21, 21
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,2,21,2} of size 336
   {2,2,21,4} of size 672
   {2,2,21,6} of size 1008
   {2,2,21,6} of size 1344
   {2,2,21,4} of size 1344
Vertex Figure Of :
   {2,2,2,21} of size 336
   {3,2,2,21} of size 504
   {4,2,2,21} of size 672
   {5,2,2,21} of size 840
   {6,2,2,21} of size 1008
   {7,2,2,21} of size 1176
   {8,2,2,21} of size 1344
   {9,2,2,21} of size 1512
   {10,2,2,21} of size 1680
   {11,2,2,21} of size 1848
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,7}*56
   7-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,2,21}*336, {2,2,42}*336
   3-fold covers : {2,2,63}*504, {2,6,21}*504, {6,2,21}*504
   4-fold covers : {8,2,21}*672, {2,2,84}*672, {2,4,42}*672a, {4,2,42}*672, {2,4,21}*672
   5-fold covers : {10,2,21}*840, {2,2,105}*840
   6-fold covers : {4,2,63}*1008, {2,2,126}*1008, {12,2,21}*1008, {4,6,21}*1008, {2,6,42}*1008b, {2,6,42}*1008c, {6,2,42}*1008
   7-fold covers : {2,2,147}*1176, {2,14,21}*1176, {14,2,21}*1176
   8-fold covers : {16,2,21}*1344, {2,4,84}*1344a, {4,2,84}*1344, {4,4,42}*1344, {2,2,168}*1344, {2,8,42}*1344, {8,2,42}*1344, {4,4,21}*1344b, {2,8,21}*1344, {2,4,42}*1344
   9-fold covers : {2,2,189}*1512, {2,6,63}*1512, {6,2,63}*1512, {18,2,21}*1512, {6,6,21}*1512a, {2,6,21}*1512, {6,6,21}*1512b
   10-fold covers : {20,2,21}*1680, {4,2,105}*1680, {2,10,42}*1680, {10,2,42}*1680, {2,2,210}*1680
   11-fold covers : {22,2,21}*1848, {2,2,231}*1848
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25);;
s3 := ( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);;
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*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(25)!(1,2);
s1 := Sym(25)!(3,4);
s2 := Sym(25)!( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)
(24,25);
s3 := Sym(25)!( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24);
poly := sub<Sym(25)|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*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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