Polytope of Type {4,57}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,57}*456
if this polytope has a name.
Group : SmallGroup(456,43)
Rank : 3
Schlafli Type : {4,57}
Number of vertices, edges, etc : 4, 114, 57
Order of s₀s₁s₂ : 57
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,57,2} of size 912
   {4,57,4} of size 1824
Vertex Figure Of :
   {2,4,57} of size 912
Quotients (Maximal Quotients in Boldface) :
   19-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,57}*912, {4,114}*912b, {4,114}*912c
   3-fold covers : {4,171}*1368
   4-fold covers : {4,228}*1824b, {4,228}*1824c, {8,57}*1824, {4,114}*1824
Permutation Representation (GAP) :

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