Polytope of Type {34,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {34,4}*272
Also Known As : {34,4|2}. if this polytope has another name.
Group : SmallGroup(272,40)
Rank : 3
Schlafli Type : {34,4}
Number of vertices, edges, etc : 34, 68, 4
Order of s₀s₁s₂ : 68
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {34,4,2} of size 544
   {34,4,4} of size 1088
   {34,4,6} of size 1632
   {34,4,3} of size 1632
Vertex Figure Of :
   {2,34,4} of size 544
   {4,34,4} of size 1088
   {6,34,4} of size 1632
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {34,2}*136
   4-fold quotients : {17,2}*68
   17-fold quotients : {2,4}*16
   34-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {68,4}*544, {34,8}*544
   3-fold covers : {34,12}*816, {102,4}*816a
   4-fold covers : {68,8}*1088a, {136,4}*1088a, {68,8}*1088b, {136,4}*1088b, {68,4}*1088, {34,16}*1088
   5-fold covers : {34,20}*1360, {170,4}*1360
   6-fold covers : {34,24}*1632, {68,12}*1632, {204,4}*1632a, {102,8}*1632
   7-fold covers : {34,28}*1904, {238,4}*1904
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope