Polytope of Type {2,34,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,34,4}*544
if this polytope has a name.
Group : SmallGroup(544,225)
Rank : 4
Schlafli Type : {2,34,4}
Number of vertices, edges, etc : 2, 34, 68, 4
Order of s₀s₁s₂s₃ : 68
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,34,4,2} of size 1088
Vertex Figure Of :
   {2,2,34,4} of size 1088
   {3,2,34,4} of size 1632
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,34,2}*272
   4-fold quotients : {2,17,2}*136
   17-fold quotients : {2,2,4}*32
   34-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,68,4}*1088, {4,34,4}*1088, {2,34,8}*1088
   3-fold covers : {2,34,12}*1632, {6,34,4}*1632, {2,102,4}*1632a
Permutation Representation (GAP) :

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