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Polytope of Type {30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*1080b
if this polytope has a name.
Group : SmallGroup(1080,337)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 90, 270, 18
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {15,6}*540
   3-fold quotients : {30,6}*360c
   5-fold quotients : {6,6}*216c
   6-fold quotients : {15,6}*180
   9-fold quotients : {30,2}*120
   10-fold quotients : {3,6}*108
   15-fold quotients : {6,6}*72c
   18-fold quotients : {15,2}*60
   27-fold quotients : {10,2}*40
   30-fold quotients : {3,6}*36
   45-fold quotients : {6,2}*24
   54-fold quotients : {5,2}*20
   90-fold quotients : {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)( 46, 91)( 47, 92)( 48, 93)
( 49,103)( 50,104)( 51,105)( 52,100)( 53,101)( 54,102)( 55, 97)( 56, 98)
( 57, 99)( 58, 94)( 59, 95)( 60, 96)( 61,121)( 62,122)( 63,123)( 64,133)
( 65,134)( 66,135)( 67,130)( 68,131)( 69,132)( 70,127)( 71,128)( 72,129)
( 73,124)( 74,125)( 75,126)( 76,106)( 77,107)( 78,108)( 79,118)( 80,119)
( 81,120)( 82,115)( 83,116)( 84,117)( 85,112)( 86,113)( 87,114)( 88,109)
( 89,110)( 90,111)(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)(181,226)
(182,227)(183,228)(184,238)(185,239)(186,240)(187,235)(188,236)(189,237)
(190,232)(191,233)(192,234)(193,229)(194,230)(195,231)(196,256)(197,257)
(198,258)(199,268)(200,269)(201,270)(202,265)(203,266)(204,267)(205,262)
(206,263)(207,264)(208,259)(209,260)(210,261)(211,241)(212,242)(213,243)
(214,253)(215,254)(216,255)(217,250)(218,251)(219,252)(220,247)(221,248)
(222,249)(223,244)(224,245)(225,246);;
s1 := (  1,245)(  2,246)(  3,244)(  4,242)(  5,243)(  6,241)(  7,254)(  8,255)
(  9,253)( 10,251)( 11,252)( 12,250)( 13,248)( 14,249)( 15,247)( 16,231)
( 17,229)( 18,230)( 19,228)( 20,226)( 21,227)( 22,240)( 23,238)( 24,239)
( 25,237)( 26,235)( 27,236)( 28,234)( 29,232)( 30,233)( 31,259)( 32,260)
( 33,261)( 34,256)( 35,257)( 36,258)( 37,268)( 38,269)( 39,270)( 40,265)
( 41,266)( 42,267)( 43,262)( 44,263)( 45,264)( 46,200)( 47,201)( 48,199)
( 49,197)( 50,198)( 51,196)( 52,209)( 53,210)( 54,208)( 55,206)( 56,207)
( 57,205)( 58,203)( 59,204)( 60,202)( 61,186)( 62,184)( 63,185)( 64,183)
( 65,181)( 66,182)( 67,195)( 68,193)( 69,194)( 70,192)( 71,190)( 72,191)
( 73,189)( 74,187)( 75,188)( 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)( 91,155)( 92,156)( 93,154)( 94,152)( 95,153)( 96,151)
( 97,164)( 98,165)( 99,163)(100,161)(101,162)(102,160)(103,158)(104,159)
(105,157)(106,141)(107,139)(108,140)(109,138)(110,136)(111,137)(112,150)
(113,148)(114,149)(115,147)(116,145)(117,146)(118,144)(119,142)(120,143)
(121,169)(122,170)(123,171)(124,166)(125,167)(126,168)(127,178)(128,179)
(129,180)(130,175)(131,176)(132,177)(133,172)(134,173)(135,174);;
s2 := (  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, 91)
( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)
( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)( 62,108)
( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)( 70,115)
( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)( 78,122)
( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)( 86,132)
( 87,131)( 88,133)( 89,135)( 90,134)(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)(181,226)(182,228)(183,227)(184,229)(185,231)
(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)(193,238)
(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)(201,245)
(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)(209,255)
(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)(217,262)
(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)(225,269);;
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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)( 46, 91)( 47, 92)
( 48, 93)( 49,103)( 50,104)( 51,105)( 52,100)( 53,101)( 54,102)( 55, 97)
( 56, 98)( 57, 99)( 58, 94)( 59, 95)( 60, 96)( 61,121)( 62,122)( 63,123)
( 64,133)( 65,134)( 66,135)( 67,130)( 68,131)( 69,132)( 70,127)( 71,128)
( 72,129)( 73,124)( 74,125)( 75,126)( 76,106)( 77,107)( 78,108)( 79,118)
( 80,119)( 81,120)( 82,115)( 83,116)( 84,117)( 85,112)( 86,113)( 87,114)
( 88,109)( 89,110)( 90,111)(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)
(181,226)(182,227)(183,228)(184,238)(185,239)(186,240)(187,235)(188,236)
(189,237)(190,232)(191,233)(192,234)(193,229)(194,230)(195,231)(196,256)
(197,257)(198,258)(199,268)(200,269)(201,270)(202,265)(203,266)(204,267)
(205,262)(206,263)(207,264)(208,259)(209,260)(210,261)(211,241)(212,242)
(213,243)(214,253)(215,254)(216,255)(217,250)(218,251)(219,252)(220,247)
(221,248)(222,249)(223,244)(224,245)(225,246);
s1 := Sym(270)!(  1,245)(  2,246)(  3,244)(  4,242)(  5,243)(  6,241)(  7,254)
(  8,255)(  9,253)( 10,251)( 11,252)( 12,250)( 13,248)( 14,249)( 15,247)
( 16,231)( 17,229)( 18,230)( 19,228)( 20,226)( 21,227)( 22,240)( 23,238)
( 24,239)( 25,237)( 26,235)( 27,236)( 28,234)( 29,232)( 30,233)( 31,259)
( 32,260)( 33,261)( 34,256)( 35,257)( 36,258)( 37,268)( 38,269)( 39,270)
( 40,265)( 41,266)( 42,267)( 43,262)( 44,263)( 45,264)( 46,200)( 47,201)
( 48,199)( 49,197)( 50,198)( 51,196)( 52,209)( 53,210)( 54,208)( 55,206)
( 56,207)( 57,205)( 58,203)( 59,204)( 60,202)( 61,186)( 62,184)( 63,185)
( 64,183)( 65,181)( 66,182)( 67,195)( 68,193)( 69,194)( 70,192)( 71,190)
( 72,191)( 73,189)( 74,187)( 75,188)( 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)( 91,155)( 92,156)( 93,154)( 94,152)( 95,153)
( 96,151)( 97,164)( 98,165)( 99,163)(100,161)(101,162)(102,160)(103,158)
(104,159)(105,157)(106,141)(107,139)(108,140)(109,138)(110,136)(111,137)
(112,150)(113,148)(114,149)(115,147)(116,145)(117,146)(118,144)(119,142)
(120,143)(121,169)(122,170)(123,171)(124,166)(125,167)(126,168)(127,178)
(128,179)(129,180)(130,175)(131,176)(132,177)(133,172)(134,173)(135,174);
s2 := 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, 91)( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)
( 54, 98)( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)
( 62,108)( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)
( 70,115)( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)
( 78,122)( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)
( 86,132)( 87,131)( 88,133)( 89,135)( 90,134)(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)(181,226)(182,228)(183,227)(184,229)
(185,231)(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)
(193,238)(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)
(201,245)(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)
(209,255)(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)
(217,262)(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)
(225,269);
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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