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Polytope of Type {30,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,18}*1080b
Also Known As : {30,18|2}. if this polytope has another name.
Group : SmallGroup(1080,336)
Rank : 3
Schlafli Type : {30,18}
Number of vertices, edges, etc : 30, 270, 18
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 : {10,18}*360, {30,6}*360b
   5-fold quotients : {6,18}*216a
   9-fold quotients : {10,6}*120, {30,2}*120
   15-fold quotients : {2,18}*72, {6,6}*72a
   18-fold quotients : {15,2}*60
   27-fold quotients : {10,2}*40
   30-fold quotients : {2,9}*36
   45-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {5,2}*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 := (  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 16, 31)( 17, 32)
( 18, 33)( 19, 43)( 20, 44)( 21, 45)( 22, 40)( 23, 41)( 24, 42)( 25, 37)
( 26, 38)( 27, 39)( 28, 34)( 29, 35)( 30, 36)( 49, 58)( 50, 59)( 51, 60)
( 52, 55)( 53, 56)( 54, 57)( 61, 76)( 62, 77)( 63, 78)( 64, 88)( 65, 89)
( 66, 90)( 67, 85)( 68, 86)( 69, 87)( 70, 82)( 71, 83)( 72, 84)( 73, 79)
( 74, 80)( 75, 81)( 94,103)( 95,104)( 96,105)( 97,100)( 98,101)( 99,102)
(106,121)(107,122)(108,123)(109,133)(110,134)(111,135)(112,130)(113,131)
(114,132)(115,127)(116,128)(117,129)(118,124)(119,125)(120,126)(139,148)
(140,149)(141,150)(142,145)(143,146)(144,147)(151,166)(152,167)(153,168)
(154,178)(155,179)(156,180)(157,175)(158,176)(159,177)(160,172)(161,173)
(162,174)(163,169)(164,170)(165,171)(184,193)(185,194)(186,195)(187,190)
(188,191)(189,192)(196,211)(197,212)(198,213)(199,223)(200,224)(201,225)
(202,220)(203,221)(204,222)(205,217)(206,218)(207,219)(208,214)(209,215)
(210,216)(229,238)(230,239)(231,240)(232,235)(233,236)(234,237)(241,256)
(242,257)(243,258)(244,268)(245,269)(246,270)(247,265)(248,266)(249,267)
(250,262)(251,263)(252,264)(253,259)(254,260)(255,261);;
s1 := (  1, 19)(  2, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)(  8, 30)
(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)( 31, 34)
( 32, 36)( 33, 35)( 37, 43)( 38, 45)( 39, 44)( 41, 42)( 46,110)( 47,109)
( 48,111)( 49,107)( 50,106)( 51,108)( 52,119)( 53,118)( 54,120)( 55,116)
( 56,115)( 57,117)( 58,113)( 59,112)( 60,114)( 61, 95)( 62, 94)( 63, 96)
( 64, 92)( 65, 91)( 66, 93)( 67,104)( 68,103)( 69,105)( 70,101)( 71,100)
( 72,102)( 73, 98)( 74, 97)( 75, 99)( 76,125)( 77,124)( 78,126)( 79,122)
( 80,121)( 81,123)( 82,134)( 83,133)( 84,135)( 85,131)( 86,130)( 87,132)
( 88,128)( 89,127)( 90,129)(136,154)(137,156)(138,155)(139,151)(140,153)
(141,152)(142,163)(143,165)(144,164)(145,160)(146,162)(147,161)(148,157)
(149,159)(150,158)(166,169)(167,171)(168,170)(172,178)(173,180)(174,179)
(176,177)(181,245)(182,244)(183,246)(184,242)(185,241)(186,243)(187,254)
(188,253)(189,255)(190,251)(191,250)(192,252)(193,248)(194,247)(195,249)
(196,230)(197,229)(198,231)(199,227)(200,226)(201,228)(202,239)(203,238)
(204,240)(205,236)(206,235)(207,237)(208,233)(209,232)(210,234)(211,260)
(212,259)(213,261)(214,257)(215,256)(216,258)(217,269)(218,268)(219,270)
(220,266)(221,265)(222,267)(223,263)(224,262)(225,264);;
s2 := (  1,181)(  2,183)(  3,182)(  4,184)(  5,186)(  6,185)(  7,187)(  8,189)
(  9,188)( 10,190)( 11,192)( 12,191)( 13,193)( 14,195)( 15,194)( 16,196)
( 17,198)( 18,197)( 19,199)( 20,201)( 21,200)( 22,202)( 23,204)( 24,203)
( 25,205)( 26,207)( 27,206)( 28,208)( 29,210)( 30,209)( 31,211)( 32,213)
( 33,212)( 34,214)( 35,216)( 36,215)( 37,217)( 38,219)( 39,218)( 40,220)
( 41,222)( 42,221)( 43,223)( 44,225)( 45,224)( 46,136)( 47,138)( 48,137)
( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)( 56,147)
( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)( 64,154)
( 65,156)( 66,155)( 67,157)( 68,159)( 69,158)( 70,160)( 71,162)( 72,161)
( 73,163)( 74,165)( 75,164)( 76,166)( 77,168)( 78,167)( 79,169)( 80,171)
( 81,170)( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)
( 89,180)( 90,179)( 91,227)( 92,226)( 93,228)( 94,230)( 95,229)( 96,231)
( 97,233)( 98,232)( 99,234)(100,236)(101,235)(102,237)(103,239)(104,238)
(105,240)(106,242)(107,241)(108,243)(109,245)(110,244)(111,246)(112,248)
(113,247)(114,249)(115,251)(116,250)(117,252)(118,254)(119,253)(120,255)
(121,257)(122,256)(123,258)(124,260)(125,259)(126,261)(127,263)(128,262)
(129,264)(130,266)(131,265)(132,267)(133,269)(134,268)(135,270);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(270)!(  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 16, 31)
( 17, 32)( 18, 33)( 19, 43)( 20, 44)( 21, 45)( 22, 40)( 23, 41)( 24, 42)
( 25, 37)( 26, 38)( 27, 39)( 28, 34)( 29, 35)( 30, 36)( 49, 58)( 50, 59)
( 51, 60)( 52, 55)( 53, 56)( 54, 57)( 61, 76)( 62, 77)( 63, 78)( 64, 88)
( 65, 89)( 66, 90)( 67, 85)( 68, 86)( 69, 87)( 70, 82)( 71, 83)( 72, 84)
( 73, 79)( 74, 80)( 75, 81)( 94,103)( 95,104)( 96,105)( 97,100)( 98,101)
( 99,102)(106,121)(107,122)(108,123)(109,133)(110,134)(111,135)(112,130)
(113,131)(114,132)(115,127)(116,128)(117,129)(118,124)(119,125)(120,126)
(139,148)(140,149)(141,150)(142,145)(143,146)(144,147)(151,166)(152,167)
(153,168)(154,178)(155,179)(156,180)(157,175)(158,176)(159,177)(160,172)
(161,173)(162,174)(163,169)(164,170)(165,171)(184,193)(185,194)(186,195)
(187,190)(188,191)(189,192)(196,211)(197,212)(198,213)(199,223)(200,224)
(201,225)(202,220)(203,221)(204,222)(205,217)(206,218)(207,219)(208,214)
(209,215)(210,216)(229,238)(230,239)(231,240)(232,235)(233,236)(234,237)
(241,256)(242,257)(243,258)(244,268)(245,269)(246,270)(247,265)(248,266)
(249,267)(250,262)(251,263)(252,264)(253,259)(254,260)(255,261);
s1 := Sym(270)!(  1, 19)(  2, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)
(  8, 30)(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)
( 31, 34)( 32, 36)( 33, 35)( 37, 43)( 38, 45)( 39, 44)( 41, 42)( 46,110)
( 47,109)( 48,111)( 49,107)( 50,106)( 51,108)( 52,119)( 53,118)( 54,120)
( 55,116)( 56,115)( 57,117)( 58,113)( 59,112)( 60,114)( 61, 95)( 62, 94)
( 63, 96)( 64, 92)( 65, 91)( 66, 93)( 67,104)( 68,103)( 69,105)( 70,101)
( 71,100)( 72,102)( 73, 98)( 74, 97)( 75, 99)( 76,125)( 77,124)( 78,126)
( 79,122)( 80,121)( 81,123)( 82,134)( 83,133)( 84,135)( 85,131)( 86,130)
( 87,132)( 88,128)( 89,127)( 90,129)(136,154)(137,156)(138,155)(139,151)
(140,153)(141,152)(142,163)(143,165)(144,164)(145,160)(146,162)(147,161)
(148,157)(149,159)(150,158)(166,169)(167,171)(168,170)(172,178)(173,180)
(174,179)(176,177)(181,245)(182,244)(183,246)(184,242)(185,241)(186,243)
(187,254)(188,253)(189,255)(190,251)(191,250)(192,252)(193,248)(194,247)
(195,249)(196,230)(197,229)(198,231)(199,227)(200,226)(201,228)(202,239)
(203,238)(204,240)(205,236)(206,235)(207,237)(208,233)(209,232)(210,234)
(211,260)(212,259)(213,261)(214,257)(215,256)(216,258)(217,269)(218,268)
(219,270)(220,266)(221,265)(222,267)(223,263)(224,262)(225,264);
s2 := Sym(270)!(  1,181)(  2,183)(  3,182)(  4,184)(  5,186)(  6,185)(  7,187)
(  8,189)(  9,188)( 10,190)( 11,192)( 12,191)( 13,193)( 14,195)( 15,194)
( 16,196)( 17,198)( 18,197)( 19,199)( 20,201)( 21,200)( 22,202)( 23,204)
( 24,203)( 25,205)( 26,207)( 27,206)( 28,208)( 29,210)( 30,209)( 31,211)
( 32,213)( 33,212)( 34,214)( 35,216)( 36,215)( 37,217)( 38,219)( 39,218)
( 40,220)( 41,222)( 42,221)( 43,223)( 44,225)( 45,224)( 46,136)( 47,138)
( 48,137)( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)
( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)
( 64,154)( 65,156)( 66,155)( 67,157)( 68,159)( 69,158)( 70,160)( 71,162)
( 72,161)( 73,163)( 74,165)( 75,164)( 76,166)( 77,168)( 78,167)( 79,169)
( 80,171)( 81,170)( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)
( 88,178)( 89,180)( 90,179)( 91,227)( 92,226)( 93,228)( 94,230)( 95,229)
( 96,231)( 97,233)( 98,232)( 99,234)(100,236)(101,235)(102,237)(103,239)
(104,238)(105,240)(106,242)(107,241)(108,243)(109,245)(110,244)(111,246)
(112,248)(113,247)(114,249)(115,251)(116,250)(117,252)(118,254)(119,253)
(120,255)(121,257)(122,256)(123,258)(124,260)(125,259)(126,261)(127,263)
(128,262)(129,264)(130,266)(131,265)(132,267)(133,269)(134,268)(135,270);
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, 
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, 
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.
to this polytope