Polytope of Type {6,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,10}*720c
if this polytope has a name.
Group : SmallGroup(720,813)
Rank : 4
Schlafli Type : {6,6,10}
Number of vertices, edges, etc : 6, 18, 30, 10
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,10,2} of size 1440
Vertex Figure Of :
   {2,6,6,10} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,10}*360
   3-fold quotients : {6,2,10}*240
   5-fold quotients : {6,6,2}*144c
   6-fold quotients : {3,2,10}*120, {6,2,5}*120
   9-fold quotients : {2,2,10}*80
   10-fold quotients : {3,6,2}*72
   12-fold quotients : {3,2,5}*60
   15-fold quotients : {6,2,2}*48
   18-fold quotients : {2,2,5}*40
   30-fold quotients : {3,2,2}*24
   45-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6,10}*1440b, {6,6,20}*1440c, {6,12,10}*1440c
Permutation Representation (GAP) :

s0 := ( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)
(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40)(51,56)
(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,86)(67,87)
(68,88)(69,89)(70,90)(71,81)(72,82)(73,83)(74,84)(75,85);;
s1 := ( 1,66)( 2,67)( 3,68)( 4,69)( 5,70)( 6,61)( 7,62)( 8,63)( 9,64)(10,65)
(11,71)(12,72)(13,73)(14,74)(15,75)(16,51)(17,52)(18,53)(19,54)(20,55)(21,46)
(22,47)(23,48)(24,49)(25,50)(26,56)(27,57)(28,58)(29,59)(30,60)(31,81)(32,82)
(33,83)(34,84)(35,85)(36,76)(37,77)(38,78)(39,79)(40,80)(41,86)(42,87)(43,88)
(44,89)(45,90);;
s2 := ( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(17,20)(18,19)(21,26)
(22,30)(23,29)(24,28)(25,27)(32,35)(33,34)(36,41)(37,45)(38,44)(39,43)(40,42)
(47,50)(48,49)(51,56)(52,60)(53,59)(54,58)(55,57)(62,65)(63,64)(66,71)(67,75)
(68,74)(69,73)(70,72)(77,80)(78,79)(81,86)(82,90)(83,89)(84,88)(85,87);;
s3 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)(51,52)
(53,55)(56,57)(58,60)(61,62)(63,65)(66,67)(68,70)(71,72)(73,75)(76,77)(78,80)
(81,82)(83,85)(86,87)(88,90);;
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*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
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(90)!( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,31)(17,32)(18,33)(19,34)
(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40)
(51,56)(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,86)
(67,87)(68,88)(69,89)(70,90)(71,81)(72,82)(73,83)(74,84)(75,85);
s1 := Sym(90)!( 1,66)( 2,67)( 3,68)( 4,69)( 5,70)( 6,61)( 7,62)( 8,63)( 9,64)
(10,65)(11,71)(12,72)(13,73)(14,74)(15,75)(16,51)(17,52)(18,53)(19,54)(20,55)
(21,46)(22,47)(23,48)(24,49)(25,50)(26,56)(27,57)(28,58)(29,59)(30,60)(31,81)
(32,82)(33,83)(34,84)(35,85)(36,76)(37,77)(38,78)(39,79)(40,80)(41,86)(42,87)
(43,88)(44,89)(45,90);
s2 := Sym(90)!( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(17,20)(18,19)
(21,26)(22,30)(23,29)(24,28)(25,27)(32,35)(33,34)(36,41)(37,45)(38,44)(39,43)
(40,42)(47,50)(48,49)(51,56)(52,60)(53,59)(54,58)(55,57)(62,65)(63,64)(66,71)
(67,75)(68,74)(69,73)(70,72)(77,80)(78,79)(81,86)(82,90)(83,89)(84,88)(85,87);
s3 := Sym(90)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)
(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)
(51,52)(53,55)(56,57)(58,60)(61,62)(63,65)(66,67)(68,70)(71,72)(73,75)(76,77)
(78,80)(81,82)(83,85)(86,87)(88,90);
poly := sub<Sym(90)|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*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope