Polytope of Type {6,90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,90}*1080a
Also Known As : {6,90|2}. if this polytope has another name.
Group : SmallGroup(1080,335)
Rank : 3
Schlafli Type : {6,90}
Number of vertices, edges, etc : 6, 270, 90
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 : {2,90}*360, {6,30}*360b
   5-fold quotients : {6,18}*216a
   6-fold quotients : {2,45}*180
   9-fold quotients : {6,10}*120, {2,30}*120
   15-fold quotients : {2,18}*72, {6,6}*72a
   18-fold quotients : {2,15}*60
   27-fold quotients : {2,10}*40
   30-fold quotients : {2,9}*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 := ( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)
( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,106)
( 62,107)( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)
( 70,115)( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 76,121)( 77,122)
( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)
( 86,131)( 87,132)( 88,133)( 89,134)( 90,135)(181,226)(182,227)(183,228)
(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)
(192,237)(193,238)(194,239)(195,240)(196,241)(197,242)(198,243)(199,244)
(200,245)(201,246)(202,247)(203,248)(204,249)(205,250)(206,251)(207,252)
(208,253)(209,254)(210,255)(211,256)(212,257)(213,258)(214,259)(215,260)
(216,261)(217,262)(218,263)(219,264)(220,265)(221,266)(222,267)(223,268)
(224,269)(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, 77)
( 17, 76)( 18, 78)( 19, 89)( 20, 88)( 21, 90)( 22, 86)( 23, 85)( 24, 87)
( 25, 83)( 26, 82)( 27, 84)( 28, 80)( 29, 79)( 30, 81)( 31, 62)( 32, 61)
( 33, 63)( 34, 74)( 35, 73)( 36, 75)( 37, 71)( 38, 70)( 39, 72)( 40, 68)
( 41, 67)( 42, 69)( 43, 65)( 44, 64)( 45, 66)( 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)(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)(151,212)(152,211)(153,213)(154,224)(155,223)(156,225)
(157,221)(158,220)(159,222)(160,218)(161,217)(162,219)(163,215)(164,214)
(165,216)(166,197)(167,196)(168,198)(169,209)(170,208)(171,210)(172,206)
(173,205)(174,207)(175,203)(176,202)(177,204)(178,200)(179,199)(180,201)
(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);;
s2 := (  1,154)(  2,156)(  3,155)(  4,151)(  5,153)(  6,152)(  7,163)(  8,165)
(  9,164)( 10,160)( 11,162)( 12,161)( 13,157)( 14,159)( 15,158)( 16,139)
( 17,141)( 18,140)( 19,136)( 20,138)( 21,137)( 22,148)( 23,150)( 24,149)
( 25,145)( 26,147)( 27,146)( 28,142)( 29,144)( 30,143)( 31,170)( 32,169)
( 33,171)( 34,167)( 35,166)( 36,168)( 37,179)( 38,178)( 39,180)( 40,176)
( 41,175)( 42,177)( 43,173)( 44,172)( 45,174)( 46,199)( 47,201)( 48,200)
( 49,196)( 50,198)( 51,197)( 52,208)( 53,210)( 54,209)( 55,205)( 56,207)
( 57,206)( 58,202)( 59,204)( 60,203)( 61,184)( 62,186)( 63,185)( 64,181)
( 65,183)( 66,182)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)( 72,191)
( 73,187)( 74,189)( 75,188)( 76,215)( 77,214)( 78,216)( 79,212)( 80,211)
( 81,213)( 82,224)( 83,223)( 84,225)( 85,221)( 86,220)( 87,222)( 88,218)
( 89,217)( 90,219)( 91,244)( 92,246)( 93,245)( 94,241)( 95,243)( 96,242)
( 97,253)( 98,255)( 99,254)(100,250)(101,252)(102,251)(103,247)(104,249)
(105,248)(106,229)(107,231)(108,230)(109,226)(110,228)(111,227)(112,238)
(113,240)(114,239)(115,235)(116,237)(117,236)(118,232)(119,234)(120,233)
(121,260)(122,259)(123,261)(124,257)(125,256)(126,258)(127,269)(128,268)
(129,270)(130,266)(131,265)(132,267)(133,263)(134,262)(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, 
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*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*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)!( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)
( 53, 98)( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)
( 61,106)( 62,107)( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)
( 69,114)( 70,115)( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 76,121)
( 77,122)( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)
( 85,130)( 86,131)( 87,132)( 88,133)( 89,134)( 90,135)(181,226)(182,227)
(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)
(191,236)(192,237)(193,238)(194,239)(195,240)(196,241)(197,242)(198,243)
(199,244)(200,245)(201,246)(202,247)(203,248)(204,249)(205,250)(206,251)
(207,252)(208,253)(209,254)(210,255)(211,256)(212,257)(213,258)(214,259)
(215,260)(216,261)(217,262)(218,263)(219,264)(220,265)(221,266)(222,267)
(223,268)(224,269)(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, 77)( 17, 76)( 18, 78)( 19, 89)( 20, 88)( 21, 90)( 22, 86)( 23, 85)
( 24, 87)( 25, 83)( 26, 82)( 27, 84)( 28, 80)( 29, 79)( 30, 81)( 31, 62)
( 32, 61)( 33, 63)( 34, 74)( 35, 73)( 36, 75)( 37, 71)( 38, 70)( 39, 72)
( 40, 68)( 41, 67)( 42, 69)( 43, 65)( 44, 64)( 45, 66)( 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)(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)(151,212)(152,211)(153,213)(154,224)(155,223)
(156,225)(157,221)(158,220)(159,222)(160,218)(161,217)(162,219)(163,215)
(164,214)(165,216)(166,197)(167,196)(168,198)(169,209)(170,208)(171,210)
(172,206)(173,205)(174,207)(175,203)(176,202)(177,204)(178,200)(179,199)
(180,201)(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);
s2 := Sym(270)!(  1,154)(  2,156)(  3,155)(  4,151)(  5,153)(  6,152)(  7,163)
(  8,165)(  9,164)( 10,160)( 11,162)( 12,161)( 13,157)( 14,159)( 15,158)
( 16,139)( 17,141)( 18,140)( 19,136)( 20,138)( 21,137)( 22,148)( 23,150)
( 24,149)( 25,145)( 26,147)( 27,146)( 28,142)( 29,144)( 30,143)( 31,170)
( 32,169)( 33,171)( 34,167)( 35,166)( 36,168)( 37,179)( 38,178)( 39,180)
( 40,176)( 41,175)( 42,177)( 43,173)( 44,172)( 45,174)( 46,199)( 47,201)
( 48,200)( 49,196)( 50,198)( 51,197)( 52,208)( 53,210)( 54,209)( 55,205)
( 56,207)( 57,206)( 58,202)( 59,204)( 60,203)( 61,184)( 62,186)( 63,185)
( 64,181)( 65,183)( 66,182)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)
( 72,191)( 73,187)( 74,189)( 75,188)( 76,215)( 77,214)( 78,216)( 79,212)
( 80,211)( 81,213)( 82,224)( 83,223)( 84,225)( 85,221)( 86,220)( 87,222)
( 88,218)( 89,217)( 90,219)( 91,244)( 92,246)( 93,245)( 94,241)( 95,243)
( 96,242)( 97,253)( 98,255)( 99,254)(100,250)(101,252)(102,251)(103,247)
(104,249)(105,248)(106,229)(107,231)(108,230)(109,226)(110,228)(111,227)
(112,238)(113,240)(114,239)(115,235)(116,237)(117,236)(118,232)(119,234)
(120,233)(121,260)(122,259)(123,261)(124,257)(125,256)(126,258)(127,269)
(128,268)(129,270)(130,266)(131,265)(132,267)(133,263)(134,262)(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, 
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*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*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