Polytope of Type {5,10,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10,6}*600
if this polytope has a name.
Group : SmallGroup(600,174)
Rank : 4
Schlafli Type : {5,10,6}
Number of vertices, edges, etc : 5, 25, 30, 6
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {5,10,6,2} of size 1200
   {5,10,6,3} of size 1800
Vertex Figure Of :
   {2,5,10,6} of size 1200
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {5,10,2}*200
   5-fold quotients : {5,2,6}*120
   10-fold quotients : {5,2,3}*60
   15-fold quotients : {5,2,2}*40
Covers (Minimal Covers in Boldface) :
   2-fold covers : {5,10,12}*1200, {10,10,6}*1200c
   3-fold covers : {5,10,18}*1800, {15,10,6}*1800
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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