Polytope of Type {4,42,2}

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

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

to this polytope