Polytope of Type {3,10,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,10,6}*1800
Also Known As : {{3,10}₆,{10,6|2}}. if this polytope has another name.
Group : SmallGroup(1800,575)
Rank : 4
Schlafli Type : {3,10,6}
Number of vertices, edges, etc : 15, 75, 150, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,10,2}*600
   25-fold quotients : {3,2,6}*72
   50-fold quotients : {3,2,3}*36
   75-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)
(27,30)(28,29)(31,32)(33,35)(36,38)(39,40)(41,44)(42,43)(46,50)(47,49)(52,55)
(53,54)(56,57)(58,60)(61,63)(64,65)(66,69)(67,68)(71,75)(72,74);;
s1 := ( 2, 7)( 3,13)( 4,19)( 5,25)( 6,21)( 9,14)(10,20)(11,16)(12,22)(18,23)
(27,32)(28,38)(29,44)(30,50)(31,46)(34,39)(35,45)(36,41)(37,47)(43,48)(52,57)
(53,63)(54,69)(55,75)(56,71)(59,64)(60,70)(61,66)(62,72)(68,73);;
s2 := ( 1,12)( 2,11)( 3,15)( 4,14)( 5,13)( 6, 7)( 8,10)(16,22)(17,21)(18,25)
(19,24)(20,23)(26,62)(27,61)(28,65)(29,64)(30,63)(31,57)(32,56)(33,60)(34,59)
(35,58)(36,52)(37,51)(38,55)(39,54)(40,53)(41,72)(42,71)(43,75)(44,74)(45,73)
(46,67)(47,66)(48,70)(49,69)(50,68);;
s3 := ( 1,26)( 2,27)( 3,28)( 4,29)( 5,30)( 6,31)( 7,32)( 8,33)( 9,34)(10,35)
(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)
(22,47)(23,48)(24,49)(25,50);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(75)!( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)
(22,24)(27,30)(28,29)(31,32)(33,35)(36,38)(39,40)(41,44)(42,43)(46,50)(47,49)
(52,55)(53,54)(56,57)(58,60)(61,63)(64,65)(66,69)(67,68)(71,75)(72,74);
s1 := Sym(75)!( 2, 7)( 3,13)( 4,19)( 5,25)( 6,21)( 9,14)(10,20)(11,16)(12,22)
(18,23)(27,32)(28,38)(29,44)(30,50)(31,46)(34,39)(35,45)(36,41)(37,47)(43,48)
(52,57)(53,63)(54,69)(55,75)(56,71)(59,64)(60,70)(61,66)(62,72)(68,73);
s2 := Sym(75)!( 1,12)( 2,11)( 3,15)( 4,14)( 5,13)( 6, 7)( 8,10)(16,22)(17,21)
(18,25)(19,24)(20,23)(26,62)(27,61)(28,65)(29,64)(30,63)(31,57)(32,56)(33,60)
(34,59)(35,58)(36,52)(37,51)(38,55)(39,54)(40,53)(41,72)(42,71)(43,75)(44,74)
(45,73)(46,67)(47,66)(48,70)(49,69)(50,68);
s3 := Sym(75)!( 1,26)( 2,27)( 3,28)( 4,29)( 5,30)( 6,31)( 7,32)( 8,33)( 9,34)
(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)
(21,46)(22,47)(23,48)(24,49)(25,50);
poly := sub<Sym(75)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 >;

References : None.
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