Polytope of Type {90,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,6}*1080b
if this polytope has a 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 : 6
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) :
   2-fold quotients : {45,6}*540
   3-fold quotients : {90,2}*360, {30,6}*360c
   5-fold quotients : {18,6}*216b
   6-fold quotients : {45,2}*180, {15,6}*180
   9-fold quotients : {30,2}*120
   10-fold quotients : {9,6}*108
   15-fold quotients : {18,2}*72, {6,6}*72c
   18-fold quotients : {15,2}*60
   27-fold quotients : {10,2}*40
   30-fold quotients : {9,2}*36, {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 := (  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)( 46, 91)( 47, 93)
( 48, 92)( 49,103)( 50,105)( 51,104)( 52,100)( 53,102)( 54,101)( 55, 97)
( 56, 99)( 57, 98)( 58, 94)( 59, 96)( 60, 95)( 61,122)( 62,121)( 63,123)
( 64,134)( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)( 71,127)
( 72,129)( 73,125)( 74,124)( 75,126)( 76,107)( 77,106)( 78,108)( 79,119)
( 80,118)( 81,120)( 82,116)( 83,115)( 84,117)( 85,113)( 86,112)( 87,114)
( 88,110)( 89,109)( 90,111)(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)(181,226)(182,228)(183,227)(184,238)(185,240)(186,239)(187,235)
(188,237)(189,236)(190,232)(191,234)(192,233)(193,229)(194,231)(195,230)
(196,257)(197,256)(198,258)(199,269)(200,268)(201,270)(202,266)(203,265)
(204,267)(205,263)(206,262)(207,264)(208,260)(209,259)(210,261)(211,242)
(212,241)(213,243)(214,254)(215,253)(216,255)(217,251)(218,250)(219,252)
(220,248)(221,247)(222,249)(223,245)(224,244)(225,246);;
s1 := (  1,199)(  2,201)(  3,200)(  4,196)(  5,198)(  6,197)(  7,208)(  8,210)
(  9,209)( 10,205)( 11,207)( 12,206)( 13,202)( 14,204)( 15,203)( 16,184)
( 17,186)( 18,185)( 19,181)( 20,183)( 21,182)( 22,193)( 23,195)( 24,194)
( 25,190)( 26,192)( 27,191)( 28,187)( 29,189)( 30,188)( 31,215)( 32,214)
( 33,216)( 34,212)( 35,211)( 36,213)( 37,224)( 38,223)( 39,225)( 40,221)
( 41,220)( 42,222)( 43,218)( 44,217)( 45,219)( 46,154)( 47,156)( 48,155)
( 49,151)( 50,153)( 51,152)( 52,163)( 53,165)( 54,164)( 55,160)( 56,162)
( 57,161)( 58,157)( 59,159)( 60,158)( 61,139)( 62,141)( 63,140)( 64,136)
( 65,138)( 66,137)( 67,148)( 68,150)( 69,149)( 70,145)( 71,147)( 72,146)
( 73,142)( 74,144)( 75,143)( 76,170)( 77,169)( 78,171)( 79,167)( 80,166)
( 81,168)( 82,179)( 83,178)( 84,180)( 85,176)( 86,175)( 87,177)( 88,173)
( 89,172)( 90,174)( 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);;
s2 := ( 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);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*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*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)( 46, 91)
( 47, 93)( 48, 92)( 49,103)( 50,105)( 51,104)( 52,100)( 53,102)( 54,101)
( 55, 97)( 56, 99)( 57, 98)( 58, 94)( 59, 96)( 60, 95)( 61,122)( 62,121)
( 63,123)( 64,134)( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)
( 71,127)( 72,129)( 73,125)( 74,124)( 75,126)( 76,107)( 77,106)( 78,108)
( 79,119)( 80,118)( 81,120)( 82,116)( 83,115)( 84,117)( 85,113)( 86,112)
( 87,114)( 88,110)( 89,109)( 90,111)(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)(181,226)(182,228)(183,227)(184,238)(185,240)(186,239)
(187,235)(188,237)(189,236)(190,232)(191,234)(192,233)(193,229)(194,231)
(195,230)(196,257)(197,256)(198,258)(199,269)(200,268)(201,270)(202,266)
(203,265)(204,267)(205,263)(206,262)(207,264)(208,260)(209,259)(210,261)
(211,242)(212,241)(213,243)(214,254)(215,253)(216,255)(217,251)(218,250)
(219,252)(220,248)(221,247)(222,249)(223,245)(224,244)(225,246);
s1 := Sym(270)!(  1,199)(  2,201)(  3,200)(  4,196)(  5,198)(  6,197)(  7,208)
(  8,210)(  9,209)( 10,205)( 11,207)( 12,206)( 13,202)( 14,204)( 15,203)
( 16,184)( 17,186)( 18,185)( 19,181)( 20,183)( 21,182)( 22,193)( 23,195)
( 24,194)( 25,190)( 26,192)( 27,191)( 28,187)( 29,189)( 30,188)( 31,215)
( 32,214)( 33,216)( 34,212)( 35,211)( 36,213)( 37,224)( 38,223)( 39,225)
( 40,221)( 41,220)( 42,222)( 43,218)( 44,217)( 45,219)( 46,154)( 47,156)
( 48,155)( 49,151)( 50,153)( 51,152)( 52,163)( 53,165)( 54,164)( 55,160)
( 56,162)( 57,161)( 58,157)( 59,159)( 60,158)( 61,139)( 62,141)( 63,140)
( 64,136)( 65,138)( 66,137)( 67,148)( 68,150)( 69,149)( 70,145)( 71,147)
( 72,146)( 73,142)( 74,144)( 75,143)( 76,170)( 77,169)( 78,171)( 79,167)
( 80,166)( 81,168)( 82,179)( 83,178)( 84,180)( 85,176)( 86,175)( 87,177)
( 88,173)( 89,172)( 90,174)( 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);
s2 := 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);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*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*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