Polytope of Type {270}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {270}*540
Also Known As : 270-gon, {270}. if this polytope has another name.
Group : SmallGroup(540,11)
Rank : 2
Schlafli Type : {270}
Number of vertices, edges, etc : 270, 270
Order of s0s1 : 270
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{270,2} of size 1080
Vertex Figure Of :
{2,270} of size 1080
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {135}*270
3-fold quotients : {90}*180
5-fold quotients : {54}*108
6-fold quotients : {45}*90
9-fold quotients : {30}*60
10-fold quotients : {27}*54
15-fold quotients : {18}*36
18-fold quotients : {15}*30
27-fold quotients : {10}*20
30-fold quotients : {9}*18
45-fold quotients : {6}*12
54-fold quotients : {5}*10
90-fold quotients : {3}*6
135-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {540}*1080
3-fold covers : {810}*1620
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 37)( 11, 39)( 12, 38)( 13, 44)
( 14, 43)( 15, 45)( 16, 41)( 17, 40)( 18, 42)( 19, 28)( 20, 30)( 21, 29)
( 22, 35)( 23, 34)( 24, 36)( 25, 32)( 26, 31)( 27, 33)( 46, 94)( 47, 96)
( 48, 95)( 49, 91)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 99)( 55,130)
( 56,132)( 57,131)( 58,127)( 59,129)( 60,128)( 61,134)( 62,133)( 63,135)
( 64,121)( 65,123)( 66,122)( 67,118)( 68,120)( 69,119)( 70,125)( 71,124)
( 72,126)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,116)
( 80,115)( 81,117)( 82,103)( 83,105)( 84,104)( 85,100)( 86,102)( 87,101)
( 88,107)( 89,106)( 90,108)(137,138)(139,143)(140,142)(141,144)(145,172)
(146,174)(147,173)(148,179)(149,178)(150,180)(151,176)(152,175)(153,177)
(154,163)(155,165)(156,164)(157,170)(158,169)(159,171)(160,167)(161,166)
(162,168)(181,229)(182,231)(183,230)(184,226)(185,228)(186,227)(187,233)
(188,232)(189,234)(190,265)(191,267)(192,266)(193,262)(194,264)(195,263)
(196,269)(197,268)(198,270)(199,256)(200,258)(201,257)(202,253)(203,255)
(204,254)(205,260)(206,259)(207,261)(208,247)(209,249)(210,248)(211,244)
(212,246)(213,245)(214,251)(215,250)(216,252)(217,238)(218,240)(219,239)
(220,235)(221,237)(222,236)(223,242)(224,241)(225,243);;
s1 := ( 1,190)( 2,192)( 3,191)( 4,197)( 5,196)( 6,198)( 7,194)( 8,193)
( 9,195)( 10,181)( 11,183)( 12,182)( 13,188)( 14,187)( 15,189)( 16,185)
( 17,184)( 18,186)( 19,217)( 20,219)( 21,218)( 22,224)( 23,223)( 24,225)
( 25,221)( 26,220)( 27,222)( 28,208)( 29,210)( 30,209)( 31,215)( 32,214)
( 33,216)( 34,212)( 35,211)( 36,213)( 37,199)( 38,201)( 39,200)( 40,206)
( 41,205)( 42,207)( 43,203)( 44,202)( 45,204)( 46,145)( 47,147)( 48,146)
( 49,152)( 50,151)( 51,153)( 52,149)( 53,148)( 54,150)( 55,136)( 56,138)
( 57,137)( 58,143)( 59,142)( 60,144)( 61,140)( 62,139)( 63,141)( 64,172)
( 65,174)( 66,173)( 67,179)( 68,178)( 69,180)( 70,176)( 71,175)( 72,177)
( 73,163)( 74,165)( 75,164)( 76,170)( 77,169)( 78,171)( 79,167)( 80,166)
( 81,168)( 82,154)( 83,156)( 84,155)( 85,161)( 86,160)( 87,162)( 88,158)
( 89,157)( 90,159)( 91,238)( 92,240)( 93,239)( 94,235)( 95,237)( 96,236)
( 97,242)( 98,241)( 99,243)(100,229)(101,231)(102,230)(103,226)(104,228)
(105,227)(106,233)(107,232)(108,234)(109,265)(110,267)(111,266)(112,262)
(113,264)(114,263)(115,269)(116,268)(117,270)(118,256)(119,258)(120,257)
(121,253)(122,255)(123,254)(124,260)(125,259)(126,261)(127,247)(128,249)
(129,248)(130,244)(131,246)(132,245)(133,251)(134,250)(135,252);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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, 8)( 5, 7)( 6, 9)( 10, 37)( 11, 39)( 12, 38)
( 13, 44)( 14, 43)( 15, 45)( 16, 41)( 17, 40)( 18, 42)( 19, 28)( 20, 30)
( 21, 29)( 22, 35)( 23, 34)( 24, 36)( 25, 32)( 26, 31)( 27, 33)( 46, 94)
( 47, 96)( 48, 95)( 49, 91)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 99)
( 55,130)( 56,132)( 57,131)( 58,127)( 59,129)( 60,128)( 61,134)( 62,133)
( 63,135)( 64,121)( 65,123)( 66,122)( 67,118)( 68,120)( 69,119)( 70,125)
( 71,124)( 72,126)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)
( 79,116)( 80,115)( 81,117)( 82,103)( 83,105)( 84,104)( 85,100)( 86,102)
( 87,101)( 88,107)( 89,106)( 90,108)(137,138)(139,143)(140,142)(141,144)
(145,172)(146,174)(147,173)(148,179)(149,178)(150,180)(151,176)(152,175)
(153,177)(154,163)(155,165)(156,164)(157,170)(158,169)(159,171)(160,167)
(161,166)(162,168)(181,229)(182,231)(183,230)(184,226)(185,228)(186,227)
(187,233)(188,232)(189,234)(190,265)(191,267)(192,266)(193,262)(194,264)
(195,263)(196,269)(197,268)(198,270)(199,256)(200,258)(201,257)(202,253)
(203,255)(204,254)(205,260)(206,259)(207,261)(208,247)(209,249)(210,248)
(211,244)(212,246)(213,245)(214,251)(215,250)(216,252)(217,238)(218,240)
(219,239)(220,235)(221,237)(222,236)(223,242)(224,241)(225,243);
s1 := Sym(270)!( 1,190)( 2,192)( 3,191)( 4,197)( 5,196)( 6,198)( 7,194)
( 8,193)( 9,195)( 10,181)( 11,183)( 12,182)( 13,188)( 14,187)( 15,189)
( 16,185)( 17,184)( 18,186)( 19,217)( 20,219)( 21,218)( 22,224)( 23,223)
( 24,225)( 25,221)( 26,220)( 27,222)( 28,208)( 29,210)( 30,209)( 31,215)
( 32,214)( 33,216)( 34,212)( 35,211)( 36,213)( 37,199)( 38,201)( 39,200)
( 40,206)( 41,205)( 42,207)( 43,203)( 44,202)( 45,204)( 46,145)( 47,147)
( 48,146)( 49,152)( 50,151)( 51,153)( 52,149)( 53,148)( 54,150)( 55,136)
( 56,138)( 57,137)( 58,143)( 59,142)( 60,144)( 61,140)( 62,139)( 63,141)
( 64,172)( 65,174)( 66,173)( 67,179)( 68,178)( 69,180)( 70,176)( 71,175)
( 72,177)( 73,163)( 74,165)( 75,164)( 76,170)( 77,169)( 78,171)( 79,167)
( 80,166)( 81,168)( 82,154)( 83,156)( 84,155)( 85,161)( 86,160)( 87,162)
( 88,158)( 89,157)( 90,159)( 91,238)( 92,240)( 93,239)( 94,235)( 95,237)
( 96,236)( 97,242)( 98,241)( 99,243)(100,229)(101,231)(102,230)(103,226)
(104,228)(105,227)(106,233)(107,232)(108,234)(109,265)(110,267)(111,266)
(112,262)(113,264)(114,263)(115,269)(116,268)(117,270)(118,256)(119,258)
(120,257)(121,253)(122,255)(123,254)(124,260)(125,259)(126,261)(127,247)
(128,249)(129,248)(130,244)(131,246)(132,245)(133,251)(134,250)(135,252);
poly := sub<Sym(270)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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