Polytope of Type {42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42}*84
Also Known As : 42-gon, {42}. if this polytope has another name.
Group : SmallGroup(84,14)
Rank : 2
Schlafli Type : {42}
Number of vertices, edges, etc : 42, 42
Order of s₀s₁ : 42
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {42,2} of size 168
   {42,4} of size 336
   {42,4} of size 336
   {42,4} of size 336
   {42,6} of size 504
   {42,6} of size 504
   {42,6} of size 504
   {42,8} of size 672
   {42,6} of size 672
   {42,4} of size 672
   {42,6} of size 756
   {42,10} of size 840
   {42,12} of size 1008
   {42,12} of size 1008
   {42,12} of size 1008
   {42,4} of size 1008
   {42,12} of size 1008
   {42,14} of size 1176
   {42,14} of size 1176
   {42,14} of size 1176
   {42,16} of size 1344
   {42,4} of size 1344
   {42,8} of size 1344
   {42,12} of size 1344
   {42,6} of size 1344
   {42,12} of size 1344
   {42,4} of size 1344
   {42,8} of size 1344
   {42,8} of size 1344
   {42,18} of size 1512
   {42,6} of size 1512
   {42,18} of size 1512
   {42,6} of size 1512
   {42,6} of size 1512
   {42,6} of size 1512
   {42,20} of size 1680
   {42,4} of size 1680
   {42,6} of size 1680
   {42,10} of size 1680
   {42,10} of size 1680
   {42,10} of size 1680
   {42,20} of size 1680
   {42,3} of size 1764
   {42,6} of size 1764
   {42,21} of size 1764
   {42,22} of size 1848
Vertex Figure Of :
   {2,42} of size 168
   {4,42} of size 336
   {4,42} of size 336
   {4,42} of size 336
   {6,42} of size 504
   {6,42} of size 504
   {6,42} of size 504
   {8,42} of size 672
   {6,42} of size 672
   {4,42} of size 672
   {6,42} of size 756
   {10,42} of size 840
   {12,42} of size 1008
   {12,42} of size 1008
   {12,42} of size 1008
   {4,42} of size 1008
   {12,42} of size 1008
   {14,42} of size 1176
   {14,42} of size 1176
   {14,42} of size 1176
   {16,42} of size 1344
   {4,42} of size 1344
   {8,42} of size 1344
   {12,42} of size 1344
   {6,42} of size 1344
   {12,42} of size 1344
   {4,42} of size 1344
   {8,42} of size 1344
   {8,42} of size 1344
   {18,42} of size 1512
   {6,42} of size 1512
   {18,42} of size 1512
   {6,42} of size 1512
   {6,42} of size 1512
   {6,42} of size 1512
   {20,42} of size 1680
   {4,42} of size 1680
   {6,42} of size 1680
   {10,42} of size 1680
   {10,42} of size 1680
   {10,42} of size 1680
   {20,42} of size 1680
   {3,42} of size 1764
   {6,42} of size 1764
   {21,42} of size 1764
   {22,42} of size 1848
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {21}*42
   3-fold quotients : {14}*28
   6-fold quotients : {7}*14
   7-fold quotients : {6}*12
   14-fold quotients : {3}*6
   21-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {84}*168
   3-fold covers : {126}*252
   4-fold covers : {168}*336
   5-fold covers : {210}*420
   6-fold covers : {252}*504
   7-fold covers : {294}*588
   8-fold covers : {336}*672
   9-fold covers : {378}*756
   10-fold covers : {420}*840
   11-fold covers : {462}*924
   12-fold covers : {504}*1008
   13-fold covers : {546}*1092
   14-fold covers : {588}*1176
   15-fold covers : {630}*1260
   16-fold covers : {672}*1344
   17-fold covers : {714}*1428
   18-fold covers : {756}*1512
   19-fold covers : {798}*1596
   20-fold covers : {840}*1680
   21-fold covers : {882}*1764
   22-fold covers : {924}*1848
   23-fold covers : {966}*1932
Permutation Representation (GAP) :

s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)
(23,26)(24,25)(27,28)(29,32)(30,31)(33,34)(35,38)(36,37)(39,42)(40,41);;
s1 := ( 1,17)( 2,11)( 3, 9)( 4,19)( 5, 7)( 6,29)( 8,13)(10,23)(12,21)(14,31)
(15,18)(16,39)(20,25)(22,35)(24,33)(26,41)(27,30)(28,40)(32,37)(34,36)
(38,42);;
poly := Group([s0,s1]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(42)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,14)(12,13)(15,16)(17,20)(18,19)
(21,22)(23,26)(24,25)(27,28)(29,32)(30,31)(33,34)(35,38)(36,37)(39,42)(40,41);
s1 := Sym(42)!( 1,17)( 2,11)( 3, 9)( 4,19)( 5, 7)( 6,29)( 8,13)(10,23)(12,21)
(14,31)(15,18)(16,39)(20,25)(22,35)(24,33)(26,41)(27,30)(28,40)(32,37)(34,36)
(38,42);
poly := sub<Sym(42)|s0,s1>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope