Polytope of Type {2,4,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,42}*672a
if this polytope has a name.
Group : SmallGroup(672,1237)
Rank : 4
Schlafli Type : {2,4,42}
Number of vertices, edges, etc : 2, 4, 84, 42
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,42,2} of size 1344
Vertex Figure Of :
   {2,2,4,42} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,42}*336
   3-fold quotients : {2,4,14}*224
   4-fold quotients : {2,2,21}*168
   6-fold quotients : {2,2,14}*112
   7-fold quotients : {2,4,6}*96a
   12-fold quotients : {2,2,7}*56
   14-fold quotients : {2,2,6}*48
   21-fold quotients : {2,4,2}*32
   28-fold quotients : {2,2,3}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,84}*1344a, {4,4,42}*1344, {2,8,42}*1344
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)
(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)
(65,86);;
s2 := ( 3,45)( 4,51)( 5,50)( 6,49)( 7,48)( 8,47)( 9,46)(10,59)(11,65)(12,64)
(13,63)(14,62)(15,61)(16,60)(17,52)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)
(24,66)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,80)(32,86)(33,85)(34,84)
(35,83)(36,82)(37,81)(38,73)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74);;
s3 := ( 3,11)( 4,10)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(17,18)(19,23)(20,22)
(24,32)(25,31)(26,37)(27,36)(28,35)(29,34)(30,33)(38,39)(40,44)(41,43)(45,53)
(46,52)(47,58)(48,57)(49,56)(50,55)(51,54)(59,60)(61,65)(62,64)(66,74)(67,73)
(68,79)(69,78)(70,77)(71,76)(72,75)(80,81)(82,86)(83,85);;
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, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
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*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(86)!(1,2);
s1 := Sym(86)!(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)
(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)
(65,86);
s2 := Sym(86)!( 3,45)( 4,51)( 5,50)( 6,49)( 7,48)( 8,47)( 9,46)(10,59)(11,65)
(12,64)(13,63)(14,62)(15,61)(16,60)(17,52)(18,58)(19,57)(20,56)(21,55)(22,54)
(23,53)(24,66)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,80)(32,86)(33,85)
(34,84)(35,83)(36,82)(37,81)(38,73)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74);
s3 := Sym(86)!( 3,11)( 4,10)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(17,18)(19,23)
(20,22)(24,32)(25,31)(26,37)(27,36)(28,35)(29,34)(30,33)(38,39)(40,44)(41,43)
(45,53)(46,52)(47,58)(48,57)(49,56)(50,55)(51,54)(59,60)(61,65)(62,64)(66,74)
(67,73)(68,79)(69,78)(70,77)(71,76)(72,75)(80,81)(82,86)(83,85);
poly := sub<Sym(86)|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, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, 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*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 >;

to this polytope