Polytope of Type {18,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,30}*1080b
Also Known As : {18,30|2}. if this polytope has another name.
Group : SmallGroup(1080,336)
Rank : 3
Schlafli Type : {18,30}
Number of vertices, edges, etc : 18, 270, 30
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,10}*360, {6,30}*360b
   5-fold quotients : {18,6}*216a
   9-fold quotients : {6,10}*120, {2,30}*120
   15-fold quotients : {18,2}*72, {6,6}*72a
   18-fold quotients : {2,15}*60
   27-fold quotients : {2,10}*40
   30-fold quotients : {9,2}*36
   45-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,5}*20
   90-fold quotients : {2,3}*12, {3,2}*12
   135-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 46, 92)
( 47, 91)( 48, 93)( 49, 95)( 50, 94)( 51, 96)( 52, 98)( 53, 97)( 54, 99)
( 55,101)( 56,100)( 57,102)( 58,104)( 59,103)( 60,105)( 61,107)( 62,106)
( 63,108)( 64,110)( 65,109)( 66,111)( 67,113)( 68,112)( 69,114)( 70,116)
( 71,115)( 72,117)( 73,119)( 74,118)( 75,120)( 76,122)( 77,121)( 78,123)
( 79,125)( 80,124)( 81,126)( 82,128)( 83,127)( 84,129)( 85,131)( 86,130)
( 87,132)( 88,134)( 89,133)( 90,135)(137,138)(140,141)(143,144)(146,147)
(149,150)(152,153)(155,156)(158,159)(161,162)(164,165)(167,168)(170,171)
(173,174)(176,177)(179,180)(181,227)(182,226)(183,228)(184,230)(185,229)
(186,231)(187,233)(188,232)(189,234)(190,236)(191,235)(192,237)(193,239)
(194,238)(195,240)(196,242)(197,241)(198,243)(199,245)(200,244)(201,246)
(202,248)(203,247)(204,249)(205,251)(206,250)(207,252)(208,254)(209,253)
(210,255)(211,257)(212,256)(213,258)(214,260)(215,259)(216,261)(217,263)
(218,262)(219,264)(220,266)(221,265)(222,267)(223,269)(224,268)(225,270);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 58)(  5, 60)(  6, 59)(  7, 55)(  8, 57)
(  9, 56)( 10, 52)( 11, 54)( 12, 53)( 13, 49)( 14, 51)( 15, 50)( 16, 76)
( 17, 78)( 18, 77)( 19, 88)( 20, 90)( 21, 89)( 22, 85)( 23, 87)( 24, 86)
( 25, 82)( 26, 84)( 27, 83)( 28, 79)( 29, 81)( 30, 80)( 31, 61)( 32, 63)
( 33, 62)( 34, 73)( 35, 75)( 36, 74)( 37, 70)( 38, 72)( 39, 71)( 40, 67)
( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 91, 92)( 94,104)( 95,103)
( 96,105)( 97,101)( 98,100)( 99,102)(106,122)(107,121)(108,123)(109,134)
(110,133)(111,135)(112,131)(113,130)(114,132)(115,128)(116,127)(117,129)
(118,125)(119,124)(120,126)(136,181)(137,183)(138,182)(139,193)(140,195)
(141,194)(142,190)(143,192)(144,191)(145,187)(146,189)(147,188)(148,184)
(149,186)(150,185)(151,211)(152,213)(153,212)(154,223)(155,225)(156,224)
(157,220)(158,222)(159,221)(160,217)(161,219)(162,218)(163,214)(164,216)
(165,215)(166,196)(167,198)(168,197)(169,208)(170,210)(171,209)(172,205)
(173,207)(174,206)(175,202)(176,204)(177,203)(178,199)(179,201)(180,200)
(226,227)(229,239)(230,238)(231,240)(232,236)(233,235)(234,237)(241,257)
(242,256)(243,258)(244,269)(245,268)(246,270)(247,266)(248,265)(249,267)
(250,263)(251,262)(252,264)(253,260)(254,259)(255,261);;
s2 := (  1,154)(  2,155)(  3,156)(  4,151)(  5,152)(  6,153)(  7,163)(  8,164)
(  9,165)( 10,160)( 11,161)( 12,162)( 13,157)( 14,158)( 15,159)( 16,139)
( 17,140)( 18,141)( 19,136)( 20,137)( 21,138)( 22,148)( 23,149)( 24,150)
( 25,145)( 26,146)( 27,147)( 28,142)( 29,143)( 30,144)( 31,169)( 32,170)
( 33,171)( 34,166)( 35,167)( 36,168)( 37,178)( 38,179)( 39,180)( 40,175)
( 41,176)( 42,177)( 43,172)( 44,173)( 45,174)( 46,199)( 47,200)( 48,201)
( 49,196)( 50,197)( 51,198)( 52,208)( 53,209)( 54,210)( 55,205)( 56,206)
( 57,207)( 58,202)( 59,203)( 60,204)( 61,184)( 62,185)( 63,186)( 64,181)
( 65,182)( 66,183)( 67,193)( 68,194)( 69,195)( 70,190)( 71,191)( 72,192)
( 73,187)( 74,188)( 75,189)( 76,214)( 77,215)( 78,216)( 79,211)( 80,212)
( 81,213)( 82,223)( 83,224)( 84,225)( 85,220)( 86,221)( 87,222)( 88,217)
( 89,218)( 90,219)( 91,244)( 92,245)( 93,246)( 94,241)( 95,242)( 96,243)
( 97,253)( 98,254)( 99,255)(100,250)(101,251)(102,252)(103,247)(104,248)
(105,249)(106,229)(107,230)(108,231)(109,226)(110,227)(111,228)(112,238)
(113,239)(114,240)(115,235)(116,236)(117,237)(118,232)(119,233)(120,234)
(121,259)(122,260)(123,261)(124,256)(125,257)(126,258)(127,268)(128,269)
(129,270)(130,265)(131,266)(132,267)(133,262)(134,263)(135,264);;
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, 
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(270)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 46, 92)( 47, 91)( 48, 93)( 49, 95)( 50, 94)( 51, 96)( 52, 98)( 53, 97)
( 54, 99)( 55,101)( 56,100)( 57,102)( 58,104)( 59,103)( 60,105)( 61,107)
( 62,106)( 63,108)( 64,110)( 65,109)( 66,111)( 67,113)( 68,112)( 69,114)
( 70,116)( 71,115)( 72,117)( 73,119)( 74,118)( 75,120)( 76,122)( 77,121)
( 78,123)( 79,125)( 80,124)( 81,126)( 82,128)( 83,127)( 84,129)( 85,131)
( 86,130)( 87,132)( 88,134)( 89,133)( 90,135)(137,138)(140,141)(143,144)
(146,147)(149,150)(152,153)(155,156)(158,159)(161,162)(164,165)(167,168)
(170,171)(173,174)(176,177)(179,180)(181,227)(182,226)(183,228)(184,230)
(185,229)(186,231)(187,233)(188,232)(189,234)(190,236)(191,235)(192,237)
(193,239)(194,238)(195,240)(196,242)(197,241)(198,243)(199,245)(200,244)
(201,246)(202,248)(203,247)(204,249)(205,251)(206,250)(207,252)(208,254)
(209,253)(210,255)(211,257)(212,256)(213,258)(214,260)(215,259)(216,261)
(217,263)(218,262)(219,264)(220,266)(221,265)(222,267)(223,269)(224,268)
(225,270);
s1 := Sym(270)!(  1, 46)(  2, 48)(  3, 47)(  4, 58)(  5, 60)(  6, 59)(  7, 55)
(  8, 57)(  9, 56)( 10, 52)( 11, 54)( 12, 53)( 13, 49)( 14, 51)( 15, 50)
( 16, 76)( 17, 78)( 18, 77)( 19, 88)( 20, 90)( 21, 89)( 22, 85)( 23, 87)
( 24, 86)( 25, 82)( 26, 84)( 27, 83)( 28, 79)( 29, 81)( 30, 80)( 31, 61)
( 32, 63)( 33, 62)( 34, 73)( 35, 75)( 36, 74)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 91, 92)( 94,104)
( 95,103)( 96,105)( 97,101)( 98,100)( 99,102)(106,122)(107,121)(108,123)
(109,134)(110,133)(111,135)(112,131)(113,130)(114,132)(115,128)(116,127)
(117,129)(118,125)(119,124)(120,126)(136,181)(137,183)(138,182)(139,193)
(140,195)(141,194)(142,190)(143,192)(144,191)(145,187)(146,189)(147,188)
(148,184)(149,186)(150,185)(151,211)(152,213)(153,212)(154,223)(155,225)
(156,224)(157,220)(158,222)(159,221)(160,217)(161,219)(162,218)(163,214)
(164,216)(165,215)(166,196)(167,198)(168,197)(169,208)(170,210)(171,209)
(172,205)(173,207)(174,206)(175,202)(176,204)(177,203)(178,199)(179,201)
(180,200)(226,227)(229,239)(230,238)(231,240)(232,236)(233,235)(234,237)
(241,257)(242,256)(243,258)(244,269)(245,268)(246,270)(247,266)(248,265)
(249,267)(250,263)(251,262)(252,264)(253,260)(254,259)(255,261);
s2 := Sym(270)!(  1,154)(  2,155)(  3,156)(  4,151)(  5,152)(  6,153)(  7,163)
(  8,164)(  9,165)( 10,160)( 11,161)( 12,162)( 13,157)( 14,158)( 15,159)
( 16,139)( 17,140)( 18,141)( 19,136)( 20,137)( 21,138)( 22,148)( 23,149)
( 24,150)( 25,145)( 26,146)( 27,147)( 28,142)( 29,143)( 30,144)( 31,169)
( 32,170)( 33,171)( 34,166)( 35,167)( 36,168)( 37,178)( 38,179)( 39,180)
( 40,175)( 41,176)( 42,177)( 43,172)( 44,173)( 45,174)( 46,199)( 47,200)
( 48,201)( 49,196)( 50,197)( 51,198)( 52,208)( 53,209)( 54,210)( 55,205)
( 56,206)( 57,207)( 58,202)( 59,203)( 60,204)( 61,184)( 62,185)( 63,186)
( 64,181)( 65,182)( 66,183)( 67,193)( 68,194)( 69,195)( 70,190)( 71,191)
( 72,192)( 73,187)( 74,188)( 75,189)( 76,214)( 77,215)( 78,216)( 79,211)
( 80,212)( 81,213)( 82,223)( 83,224)( 84,225)( 85,220)( 86,221)( 87,222)
( 88,217)( 89,218)( 90,219)( 91,244)( 92,245)( 93,246)( 94,241)( 95,242)
( 96,243)( 97,253)( 98,254)( 99,255)(100,250)(101,251)(102,252)(103,247)
(104,248)(105,249)(106,229)(107,230)(108,231)(109,226)(110,227)(111,228)
(112,238)(113,239)(114,240)(115,235)(116,236)(117,237)(118,232)(119,233)
(120,234)(121,259)(122,260)(123,261)(124,256)(125,257)(126,258)(127,268)
(128,269)(129,270)(130,265)(131,266)(132,267)(133,262)(134,263)(135,264);
poly := sub<Sym(270)|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, 
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope