Polytope of Type {6,10,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10,5}*600
if this polytope has a name.
Group : SmallGroup(600,174)
Rank : 4
Schlafli Type : {6,10,5}
Number of vertices, edges, etc : 6, 30, 25, 5
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,10,5,2} of size 1200
Vertex Figure Of :
   {2,6,10,5} of size 1200
   {3,6,10,5} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,10,5}*200
   5-fold quotients : {6,2,5}*120
   10-fold quotients : {3,2,5}*60
   15-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,10,5}*1200, {6,10,10}*1200b
   3-fold covers : {18,10,5}*1800, {6,10,15}*1800
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)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)
(11,36)(12,40)(13,39)(14,38)(15,37)(16,41)(17,45)(18,44)(19,43)(20,42)(21,46)
(22,50)(23,49)(24,48)(25,47)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)
(68,69)(72,75)(73,74);;
s2 := ( 1, 2)( 3, 5)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)
(14,19)(15,18)(26,27)(28,30)(31,47)(32,46)(33,50)(34,49)(35,48)(36,42)(37,41)
(38,45)(39,44)(40,43)(51,52)(53,55)(56,72)(57,71)(58,75)(59,74)(60,73)(61,67)
(62,66)(63,70)(64,69)(65,68);;
s3 := ( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,21)(12,25)(13,24)(14,23)(15,22)
(17,20)(18,19)(26,31)(27,35)(28,34)(29,33)(30,32)(36,46)(37,50)(38,49)(39,48)
(40,47)(42,45)(43,44)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71)(62,75)(63,74)
(64,73)(65,72)(67,70)(68,69);;
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*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
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)( 6,31)( 7,35)( 8,34)( 9,33)
(10,32)(11,36)(12,40)(13,39)(14,38)(15,37)(16,41)(17,45)(18,44)(19,43)(20,42)
(21,46)(22,50)(23,49)(24,48)(25,47)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)
(67,70)(68,69)(72,75)(73,74);
s2 := Sym(75)!( 1, 2)( 3, 5)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)
(13,20)(14,19)(15,18)(26,27)(28,30)(31,47)(32,46)(33,50)(34,49)(35,48)(36,42)
(37,41)(38,45)(39,44)(40,43)(51,52)(53,55)(56,72)(57,71)(58,75)(59,74)(60,73)
(61,67)(62,66)(63,70)(64,69)(65,68);
s3 := Sym(75)!( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,21)(12,25)(13,24)(14,23)
(15,22)(17,20)(18,19)(26,31)(27,35)(28,34)(29,33)(30,32)(36,46)(37,50)(38,49)
(39,48)(40,47)(42,45)(43,44)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71)(62,75)
(63,74)(64,73)(65,72)(67,70)(68,69);
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*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope