Polytope of Type {30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*360b
Also Known As : {30,6|2}. if this polytope has another name.
Group : SmallGroup(360,154)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 30, 90, 6
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {30,6,2} of size 720
   {30,6,3} of size 1080
   {30,6,4} of size 1440
   {30,6,3} of size 1440
   {30,6,4} of size 1440
Vertex Figure Of :
   {2,30,6} of size 720
   {4,30,6} of size 1440
   {4,30,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,6}*120, {30,2}*120
   5-fold quotients : {6,6}*72a
   6-fold quotients : {15,2}*60
   9-fold quotients : {10,2}*40
   15-fold quotients : {2,6}*24, {6,2}*24
   18-fold quotients : {5,2}*20
   30-fold quotients : {2,3}*12, {3,2}*12
   45-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {30,12}*720b, {60,6}*720b
   3-fold covers : {90,6}*1080a, {30,18}*1080b, {30,6}*1080c, {30,6}*1080d
   4-fold covers : {30,24}*1440b, {120,6}*1440b, {60,12}*1440b, {30,12}*1440a, {60,6}*1440d
   5-fold covers : {150,6}*1800b, {30,30}*1800f, {30,30}*1800g
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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