Polytope of Type {42,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,4}*336a
Also Known As : {42,4|2}. if this polytope has another name.
Group : SmallGroup(336,198)
Rank : 3
Schlafli Type : {42,4}
Number of vertices, edges, etc : 42, 84, 4
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 :
   {42,4,2} of size 672
   {42,4,4} of size 1344
Vertex Figure Of :
   {2,42,4} of size 672
   {4,42,4} of size 1344
   {4,42,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,2}*168
   3-fold quotients : {14,4}*112
   4-fold quotients : {21,2}*84
   6-fold quotients : {14,2}*56
   7-fold quotients : {6,4}*48a
   12-fold quotients : {7,2}*28
   14-fold quotients : {6,2}*24
   21-fold quotients : {2,4}*16
   28-fold quotients : {3,2}*12
   42-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {84,4}*672a, {42,8}*672
   3-fold covers : {126,4}*1008a, {42,12}*1008b, {42,12}*1008c
   4-fold covers : {168,4}*1344a, {84,4}*1344a, {168,4}*1344b, {84,8}*1344a, {84,8}*1344b, {42,16}*1344, {42,4}*1344b
   5-fold covers : {42,20}*1680a, {210,4}*1680a
Permutation Representation (GAP) :

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