Polytope of Type {57,4}

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

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