Polytope of Type {4,21,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,21,6}*1008
if this polytope has a name.
Group : SmallGroup(1008,903)
Rank : 4
Schlafli Type : {4,21,6}
Number of vertices, edges, etc : 4, 42, 63, 6
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,21,2}*336
   7-fold quotients : {4,3,6}*144
   21-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)
(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)
(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)
(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84);;
s1 := ( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)(13,17)
(14,18)(15,20)(16,19)(29,57)(30,58)(31,60)(32,59)(33,81)(34,82)(35,84)(36,83)
(37,77)(38,78)(39,80)(40,79)(41,73)(42,74)(43,76)(44,75)(45,69)(46,70)(47,72)
(48,71)(49,65)(50,66)(51,68)(52,67)(53,61)(54,62)(55,64)(56,63);;
s2 := ( 1,33)( 2,36)( 3,35)( 4,34)( 5,29)( 6,32)( 7,31)( 8,30)( 9,53)(10,56)
(11,55)(12,54)(13,49)(14,52)(15,51)(16,50)(17,45)(18,48)(19,47)(20,46)(21,41)
(22,44)(23,43)(24,42)(25,37)(26,40)(27,39)(28,38)(57,61)(58,64)(59,63)(60,62)
(65,81)(66,84)(67,83)(68,82)(69,77)(70,80)(71,79)(72,78)(74,76);;
s3 := (29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)
(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)
(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84);;
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, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(84)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)
(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)
(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)
(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84);
s1 := Sym(84)!( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)
(13,17)(14,18)(15,20)(16,19)(29,57)(30,58)(31,60)(32,59)(33,81)(34,82)(35,84)
(36,83)(37,77)(38,78)(39,80)(40,79)(41,73)(42,74)(43,76)(44,75)(45,69)(46,70)
(47,72)(48,71)(49,65)(50,66)(51,68)(52,67)(53,61)(54,62)(55,64)(56,63);
s2 := Sym(84)!( 1,33)( 2,36)( 3,35)( 4,34)( 5,29)( 6,32)( 7,31)( 8,30)( 9,53)
(10,56)(11,55)(12,54)(13,49)(14,52)(15,51)(16,50)(17,45)(18,48)(19,47)(20,46)
(21,41)(22,44)(23,43)(24,42)(25,37)(26,40)(27,39)(28,38)(57,61)(58,64)(59,63)
(60,62)(65,81)(66,84)(67,83)(68,82)(69,77)(70,80)(71,79)(72,78)(74,76);
s3 := Sym(84)!(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)
(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)
(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84);
poly := sub<Sym(84)|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, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope