Polytope of Type {4,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,42}*336a
Also Known As : {4,42|2}. if this polytope has another name.
Group : SmallGroup(336,198)
Rank : 3
Schlafli Type : {4,42}
Number of vertices, edges, etc : 4, 84, 42
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,42,2} of size 672
   {4,42,4} of size 1344
   {4,42,4} of size 1344
Vertex Figure Of :
   {2,4,42} of size 672
   {4,4,42} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,42}*168
   3-fold quotients : {4,14}*112
   4-fold quotients : {2,21}*84
   6-fold quotients : {2,14}*56
   7-fold quotients : {4,6}*48a
   12-fold quotients : {2,7}*28
   14-fold quotients : {2,6}*24
   21-fold quotients : {4,2}*16
   28-fold quotients : {2,3}*12
   42-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,84}*672a, {8,42}*672
   3-fold covers : {4,126}*1008a, {12,42}*1008b, {12,42}*1008c
   4-fold covers : {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b, {16,42}*1344, {4,42}*1344b
   5-fold covers : {20,42}*1680a, {4,210}*1680a
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);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, 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(84)!(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(84)!( 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(84)!( 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);
poly := sub<Sym(84)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, 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 >;

References : None.
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