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Polytope of Type {6,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,10}*1800
if this polytope has a name.
Group : SmallGroup(1800,575)
Rank : 4
Schlafli Type : {6,6,10}
Number of vertices, edges, etc : 6, 45, 75, 25
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Non-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 : {2,6,10}*600
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)
(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)
(47,72)(48,73)(49,74)(50,75);;
s1 := ( 1,26)( 2,30)( 3,29)( 4,28)( 5,27)( 6,32)( 7,31)( 8,35)( 9,34)(10,33)
(11,38)(12,37)(13,36)(14,40)(15,39)(16,44)(17,43)(18,42)(19,41)(20,45)(21,50)
(22,49)(23,48)(24,47)(25,46)(52,55)(53,54)(56,57)(58,60)(61,63)(64,65)(66,69)
(67,68)(71,75)(72,74);;
s2 := ( 2,25)( 3,19)( 4,13)( 5, 7)( 8,24)( 9,18)(10,12)(14,23)(15,17)(20,22)
(26,51)(27,75)(28,69)(29,63)(30,57)(31,56)(32,55)(33,74)(34,68)(35,62)(36,61)
(37,60)(38,54)(39,73)(40,67)(41,66)(42,65)(43,59)(44,53)(45,72)(46,71)(47,70)
(48,64)(49,58)(50,52);;
s3 := ( 1,12)( 2,11)( 3,15)( 4,14)( 5,13)( 6, 7)( 8,10)(16,22)(17,21)(18,25)
(19,24)(20,23)(26,37)(27,36)(28,40)(29,39)(30,38)(31,32)(33,35)(41,47)(42,46)
(43,50)(44,49)(45,48)(51,62)(52,61)(53,65)(54,64)(55,63)(56,57)(58,60)(66,72)
(67,71)(68,75)(69,74)(70,73);;
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, s1*s2*s3*s1*s2*s3*s1*s2*s3, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(75)!(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)
(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)
(46,71)(47,72)(48,73)(49,74)(50,75);
s1 := Sym(75)!( 1,26)( 2,30)( 3,29)( 4,28)( 5,27)( 6,32)( 7,31)( 8,35)( 9,34)
(10,33)(11,38)(12,37)(13,36)(14,40)(15,39)(16,44)(17,43)(18,42)(19,41)(20,45)
(21,50)(22,49)(23,48)(24,47)(25,46)(52,55)(53,54)(56,57)(58,60)(61,63)(64,65)
(66,69)(67,68)(71,75)(72,74);
s2 := Sym(75)!( 2,25)( 3,19)( 4,13)( 5, 7)( 8,24)( 9,18)(10,12)(14,23)(15,17)
(20,22)(26,51)(27,75)(28,69)(29,63)(30,57)(31,56)(32,55)(33,74)(34,68)(35,62)
(36,61)(37,60)(38,54)(39,73)(40,67)(41,66)(42,65)(43,59)(44,53)(45,72)(46,71)
(47,70)(48,64)(49,58)(50,52);
s3 := 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)(26,37)(27,36)(28,40)(29,39)(30,38)(31,32)(33,35)(41,47)
(42,46)(43,50)(44,49)(45,48)(51,62)(52,61)(53,65)(54,64)(55,63)(56,57)(58,60)
(66,72)(67,71)(68,75)(69,74)(70,73);
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, 
s1*s2*s3*s1*s2*s3*s1*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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