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Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1800a
if this polytope has a name.
Group : SmallGroup(1800,575)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 30, 450, 150
Order of s0s1s2 : 6
Order of s0s1s2s1 : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,10}*600b
   6-fold quotients : {3,10}*300
   25-fold quotients : {6,6}*72a
   75-fold quotients : {2,6}*24, {6,2}*24
   150-fold quotients : {2,3}*12, {3,2}*12
   225-fold quotients : {2,2}*8
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)
(26,51)(27,57)(28,63)(29,69)(30,75)(31,71)(32,52)(33,58)(34,64)(35,70)(36,66)
(37,72)(38,53)(39,59)(40,65)(41,61)(42,67)(43,73)(44,54)(45,60)(46,56)(47,62)
(48,68)(49,74)(50,55);;
s2 := ( 1,37)( 2,36)( 3,40)( 4,39)( 5,38)( 6,32)( 7,31)( 8,35)( 9,34)(10,33)
(11,27)(12,26)(13,30)(14,29)(15,28)(16,47)(17,46)(18,50)(19,49)(20,48)(21,42)
(22,41)(23,45)(24,44)(25,43)(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]);;
 
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*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 ];;
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)(26,51)(27,57)(28,63)(29,69)(30,75)(31,71)(32,52)(33,58)(34,64)(35,70)
(36,66)(37,72)(38,53)(39,59)(40,65)(41,61)(42,67)(43,73)(44,54)(45,60)(46,56)
(47,62)(48,68)(49,74)(50,55);
s2 := Sym(75)!( 1,37)( 2,36)( 3,40)( 4,39)( 5,38)( 6,32)( 7,31)( 8,35)( 9,34)
(10,33)(11,27)(12,26)(13,30)(14,29)(15,28)(16,47)(17,46)(18,50)(19,49)(20,48)
(21,42)(22,41)(23,45)(24,44)(25,43)(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>;
 
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*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 >; 
 
References : None.
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