Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*900
Also Known As : {3,6}_(5,5). if this polytope has another name.
Group : SmallGroup(900,95)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 75, 225, 150
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,6,2} of size 1800
Vertex Figure Of :
   {2,3,6} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*300
   25-fold quotients : {3,6}*36
   75-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6}*1800d
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope