Polytope of Type {90,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,6}*1080a
Also Known As : {90,6|2}. if this polytope has another name.
Group : SmallGroup(1080,335)
Rank : 3
Schlafli Type : {90,6}
Number of vertices, edges, etc : 90, 270, 6
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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) :
   3-fold quotients : {90,2}*360, {30,6}*360b
   5-fold quotients : {18,6}*216a
   6-fold quotients : {45,2}*180
   9-fold quotients : {10,6}*120, {30,2}*120
   15-fold quotients : {18,2}*72, {6,6}*72a
   18-fold quotients : {15,2}*60
   27-fold quotients : {10,2}*40
   30-fold quotients : {9,2}*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 := (  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 16, 32)
( 17, 31)( 18, 33)( 19, 44)( 20, 43)( 21, 45)( 22, 41)( 23, 40)( 24, 42)
( 25, 38)( 26, 37)( 27, 39)( 28, 35)( 29, 34)( 30, 36)( 47, 48)( 49, 58)
( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 77)( 62, 76)( 63, 78)
( 64, 89)( 65, 88)( 66, 90)( 67, 86)( 68, 85)( 69, 87)( 70, 83)( 71, 82)
( 72, 84)( 73, 80)( 74, 79)( 75, 81)( 92, 93)( 94,103)( 95,105)( 96,104)
( 97,100)( 98,102)( 99,101)(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)(137,138)(139,148)(140,150)(141,149)(142,145)(143,147)
(144,146)(151,167)(152,166)(153,168)(154,179)(155,178)(156,180)(157,176)
(158,175)(159,177)(160,173)(161,172)(162,174)(163,170)(164,169)(165,171)
(182,183)(184,193)(185,195)(186,194)(187,190)(188,192)(189,191)(196,212)
(197,211)(198,213)(199,224)(200,223)(201,225)(202,221)(203,220)(204,222)
(205,218)(206,217)(207,219)(208,215)(209,214)(210,216)(227,228)(229,238)
(230,240)(231,239)(232,235)(233,237)(234,236)(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);;
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, 35)
( 32, 34)( 33, 36)( 37, 44)( 38, 43)( 39, 45)( 40, 41)( 46,109)( 47,111)
( 48,110)( 49,106)( 50,108)( 51,107)( 52,118)( 53,120)( 54,119)( 55,115)
( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61, 94)( 62, 96)( 63, 95)
( 64, 91)( 65, 93)( 66, 92)( 67,103)( 68,105)( 69,104)( 70,100)( 71,102)
( 72,101)( 73, 97)( 74, 99)( 75, 98)( 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)(166,170)(167,169)(168,171)(172,179)(173,178)(174,180)
(175,176)(181,244)(182,246)(183,245)(184,241)(185,243)(186,242)(187,253)
(188,255)(189,254)(190,250)(191,252)(192,251)(193,247)(194,249)(195,248)
(196,229)(197,231)(198,230)(199,226)(200,228)(201,227)(202,238)(203,240)
(204,239)(205,235)(206,237)(207,236)(208,232)(209,234)(210,233)(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)(  2,182)(  3,183)(  4,184)(  5,185)(  6,186)(  7,187)(  8,188)
(  9,189)( 10,190)( 11,191)( 12,192)( 13,193)( 14,194)( 15,195)( 16,196)
( 17,197)( 18,198)( 19,199)( 20,200)( 21,201)( 22,202)( 23,203)( 24,204)
( 25,205)( 26,206)( 27,207)( 28,208)( 29,209)( 30,210)( 31,211)( 32,212)
( 33,213)( 34,214)( 35,215)( 36,216)( 37,217)( 38,218)( 39,219)( 40,220)
( 41,221)( 42,222)( 43,223)( 44,224)( 45,225)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)
( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)
( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)( 96,231)
( 97,232)( 98,233)( 99,234)(100,235)(101,236)(102,237)(103,238)(104,239)
(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)
(113,248)(114,249)(115,250)(116,251)(117,252)(118,253)(119,254)(120,255)
(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,262)(128,263)
(129,264)(130,265)(131,266)(132,267)(133,268)(134,269)(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, 
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*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*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)!(  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)
( 16, 32)( 17, 31)( 18, 33)( 19, 44)( 20, 43)( 21, 45)( 22, 41)( 23, 40)
( 24, 42)( 25, 38)( 26, 37)( 27, 39)( 28, 35)( 29, 34)( 30, 36)( 47, 48)
( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 77)( 62, 76)
( 63, 78)( 64, 89)( 65, 88)( 66, 90)( 67, 86)( 68, 85)( 69, 87)( 70, 83)
( 71, 82)( 72, 84)( 73, 80)( 74, 79)( 75, 81)( 92, 93)( 94,103)( 95,105)
( 96,104)( 97,100)( 98,102)( 99,101)(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)(137,138)(139,148)(140,150)(141,149)(142,145)
(143,147)(144,146)(151,167)(152,166)(153,168)(154,179)(155,178)(156,180)
(157,176)(158,175)(159,177)(160,173)(161,172)(162,174)(163,170)(164,169)
(165,171)(182,183)(184,193)(185,195)(186,194)(187,190)(188,192)(189,191)
(196,212)(197,211)(198,213)(199,224)(200,223)(201,225)(202,221)(203,220)
(204,222)(205,218)(206,217)(207,219)(208,215)(209,214)(210,216)(227,228)
(229,238)(230,240)(231,239)(232,235)(233,237)(234,236)(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);
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, 35)( 32, 34)( 33, 36)( 37, 44)( 38, 43)( 39, 45)( 40, 41)( 46,109)
( 47,111)( 48,110)( 49,106)( 50,108)( 51,107)( 52,118)( 53,120)( 54,119)
( 55,115)( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61, 94)( 62, 96)
( 63, 95)( 64, 91)( 65, 93)( 66, 92)( 67,103)( 68,105)( 69,104)( 70,100)
( 71,102)( 72,101)( 73, 97)( 74, 99)( 75, 98)( 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)(166,170)(167,169)(168,171)(172,179)(173,178)
(174,180)(175,176)(181,244)(182,246)(183,245)(184,241)(185,243)(186,242)
(187,253)(188,255)(189,254)(190,250)(191,252)(192,251)(193,247)(194,249)
(195,248)(196,229)(197,231)(198,230)(199,226)(200,228)(201,227)(202,238)
(203,240)(204,239)(205,235)(206,237)(207,236)(208,232)(209,234)(210,233)
(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)(  2,182)(  3,183)(  4,184)(  5,185)(  6,186)(  7,187)
(  8,188)(  9,189)( 10,190)( 11,191)( 12,192)( 13,193)( 14,194)( 15,195)
( 16,196)( 17,197)( 18,198)( 19,199)( 20,200)( 21,201)( 22,202)( 23,203)
( 24,204)( 25,205)( 26,206)( 27,207)( 28,208)( 29,209)( 30,210)( 31,211)
( 32,212)( 33,213)( 34,214)( 35,215)( 36,216)( 37,217)( 38,218)( 39,219)
( 40,220)( 41,221)( 42,222)( 43,223)( 44,224)( 45,225)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)
( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)
( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)
( 96,231)( 97,232)( 98,233)( 99,234)(100,235)(101,236)(102,237)(103,238)
(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)
(112,247)(113,248)(114,249)(115,250)(116,251)(117,252)(118,253)(119,254)
(120,255)(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,262)
(128,263)(129,264)(130,265)(131,266)(132,267)(133,268)(134,269)(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, 
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*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*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