Polytope of Type {20,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4}*1440
if this polytope has a name.
Group : SmallGroup(1440,4764)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 180, 360, 36
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,4}*720
5-fold quotients : {4,4}*288
9-fold quotients : {20,4}*160
10-fold quotients : {4,4}*144
18-fold quotients : {20,2}*80, {10,4}*80
20-fold quotients : {4,4}*72
36-fold quotients : {10,2}*40
45-fold quotients : {4,4}*32
72-fold quotients : {5,2}*20
90-fold quotients : {2,4}*16, {4,2}*16
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 4)( 6, 36)( 7, 40)( 8, 39)( 9, 38)( 10, 37)( 11, 26)
( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 16, 31)( 17, 35)( 18, 34)( 19, 33)
( 20, 32)( 22, 25)( 23, 24)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 51, 81)
( 52, 85)( 53, 84)( 54, 83)( 55, 82)( 56, 71)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)( 67, 70)( 68, 69)
( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 96,126)( 97,130)( 98,129)( 99,128)
(100,127)(101,116)(102,120)(103,119)(104,118)(105,117)(106,121)(107,125)
(108,124)(109,123)(110,122)(112,115)(113,114)(132,135)(133,134)(137,140)
(138,139)(141,171)(142,175)(143,174)(144,173)(145,172)(146,161)(147,165)
(148,164)(149,163)(150,162)(151,166)(152,170)(153,169)(154,168)(155,167)
(157,160)(158,159)(177,180)(178,179)(181,316)(182,320)(183,319)(184,318)
(185,317)(186,351)(187,355)(188,354)(189,353)(190,352)(191,341)(192,345)
(193,344)(194,343)(195,342)(196,346)(197,350)(198,349)(199,348)(200,347)
(201,336)(202,340)(203,339)(204,338)(205,337)(206,326)(207,330)(208,329)
(209,328)(210,327)(211,331)(212,335)(213,334)(214,333)(215,332)(216,321)
(217,325)(218,324)(219,323)(220,322)(221,356)(222,360)(223,359)(224,358)
(225,357)(226,271)(227,275)(228,274)(229,273)(230,272)(231,306)(232,310)
(233,309)(234,308)(235,307)(236,296)(237,300)(238,299)(239,298)(240,297)
(241,301)(242,305)(243,304)(244,303)(245,302)(246,291)(247,295)(248,294)
(249,293)(250,292)(251,281)(252,285)(253,284)(254,283)(255,282)(256,286)
(257,290)(258,289)(259,288)(260,287)(261,276)(262,280)(263,279)(264,278)
(265,277)(266,311)(267,315)(268,314)(269,313)(270,312);;
s1 := ( 1,182)( 2,181)( 3,185)( 4,184)( 5,183)( 6,187)( 7,186)( 8,190)
( 9,189)( 10,188)( 11,192)( 12,191)( 13,195)( 14,194)( 15,193)( 16,222)
( 17,221)( 18,225)( 19,224)( 20,223)( 21,212)( 22,211)( 23,215)( 24,214)
( 25,213)( 26,217)( 27,216)( 28,220)( 29,219)( 30,218)( 31,202)( 32,201)
( 33,205)( 34,204)( 35,203)( 36,207)( 37,206)( 38,210)( 39,209)( 40,208)
( 41,197)( 42,196)( 43,200)( 44,199)( 45,198)( 46,227)( 47,226)( 48,230)
( 49,229)( 50,228)( 51,232)( 52,231)( 53,235)( 54,234)( 55,233)( 56,237)
( 57,236)( 58,240)( 59,239)( 60,238)( 61,267)( 62,266)( 63,270)( 64,269)
( 65,268)( 66,257)( 67,256)( 68,260)( 69,259)( 70,258)( 71,262)( 72,261)
( 73,265)( 74,264)( 75,263)( 76,247)( 77,246)( 78,250)( 79,249)( 80,248)
( 81,252)( 82,251)( 83,255)( 84,254)( 85,253)( 86,242)( 87,241)( 88,245)
( 89,244)( 90,243)( 91,272)( 92,271)( 93,275)( 94,274)( 95,273)( 96,277)
( 97,276)( 98,280)( 99,279)(100,278)(101,282)(102,281)(103,285)(104,284)
(105,283)(106,312)(107,311)(108,315)(109,314)(110,313)(111,302)(112,301)
(113,305)(114,304)(115,303)(116,307)(117,306)(118,310)(119,309)(120,308)
(121,292)(122,291)(123,295)(124,294)(125,293)(126,297)(127,296)(128,300)
(129,299)(130,298)(131,287)(132,286)(133,290)(134,289)(135,288)(136,317)
(137,316)(138,320)(139,319)(140,318)(141,322)(142,321)(143,325)(144,324)
(145,323)(146,327)(147,326)(148,330)(149,329)(150,328)(151,357)(152,356)
(153,360)(154,359)(155,358)(156,347)(157,346)(158,350)(159,349)(160,348)
(161,352)(162,351)(163,355)(164,354)(165,353)(166,337)(167,336)(168,340)
(169,339)(170,338)(171,342)(172,341)(173,345)(174,344)(175,343)(176,332)
(177,331)(178,335)(179,334)(180,333);;
s2 := ( 1,111)( 2,112)( 3,113)( 4,114)( 5,115)( 6,121)( 7,122)( 8,123)
( 9,124)( 10,125)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,126)
( 17,127)( 18,128)( 19,129)( 20,130)( 21, 91)( 22, 92)( 23, 93)( 24, 94)
( 25, 95)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31, 96)( 32, 97)
( 33, 98)( 34, 99)( 35,100)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,156)( 47,157)( 48,158)
( 49,159)( 50,160)( 51,166)( 52,167)( 53,168)( 54,169)( 55,170)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)( 64,174)
( 65,175)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,161)( 72,162)
( 73,163)( 74,164)( 75,165)( 76,141)( 77,142)( 78,143)( 79,144)( 80,145)
( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180)(181,201)(182,202)(183,203)(184,204)(185,205)(186,211)
(187,212)(188,213)(189,214)(190,215)(196,216)(197,217)(198,218)(199,219)
(200,220)(226,246)(227,247)(228,248)(229,249)(230,250)(231,256)(232,257)
(233,258)(234,259)(235,260)(241,261)(242,262)(243,263)(244,264)(245,265)
(271,291)(272,292)(273,293)(274,294)(275,295)(276,301)(277,302)(278,303)
(279,304)(280,305)(286,306)(287,307)(288,308)(289,309)(290,310)(316,336)
(317,337)(318,338)(319,339)(320,340)(321,346)(322,347)(323,348)(324,349)
(325,350)(331,351)(332,352)(333,353)(334,354)(335,355);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(360)!( 2, 5)( 3, 4)( 6, 36)( 7, 40)( 8, 39)( 9, 38)( 10, 37)
( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 16, 31)( 17, 35)( 18, 34)
( 19, 33)( 20, 32)( 22, 25)( 23, 24)( 42, 45)( 43, 44)( 47, 50)( 48, 49)
( 51, 81)( 52, 85)( 53, 84)( 54, 83)( 55, 82)( 56, 71)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)( 67, 70)
( 68, 69)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 96,126)( 97,130)( 98,129)
( 99,128)(100,127)(101,116)(102,120)(103,119)(104,118)(105,117)(106,121)
(107,125)(108,124)(109,123)(110,122)(112,115)(113,114)(132,135)(133,134)
(137,140)(138,139)(141,171)(142,175)(143,174)(144,173)(145,172)(146,161)
(147,165)(148,164)(149,163)(150,162)(151,166)(152,170)(153,169)(154,168)
(155,167)(157,160)(158,159)(177,180)(178,179)(181,316)(182,320)(183,319)
(184,318)(185,317)(186,351)(187,355)(188,354)(189,353)(190,352)(191,341)
(192,345)(193,344)(194,343)(195,342)(196,346)(197,350)(198,349)(199,348)
(200,347)(201,336)(202,340)(203,339)(204,338)(205,337)(206,326)(207,330)
(208,329)(209,328)(210,327)(211,331)(212,335)(213,334)(214,333)(215,332)
(216,321)(217,325)(218,324)(219,323)(220,322)(221,356)(222,360)(223,359)
(224,358)(225,357)(226,271)(227,275)(228,274)(229,273)(230,272)(231,306)
(232,310)(233,309)(234,308)(235,307)(236,296)(237,300)(238,299)(239,298)
(240,297)(241,301)(242,305)(243,304)(244,303)(245,302)(246,291)(247,295)
(248,294)(249,293)(250,292)(251,281)(252,285)(253,284)(254,283)(255,282)
(256,286)(257,290)(258,289)(259,288)(260,287)(261,276)(262,280)(263,279)
(264,278)(265,277)(266,311)(267,315)(268,314)(269,313)(270,312);
s1 := Sym(360)!( 1,182)( 2,181)( 3,185)( 4,184)( 5,183)( 6,187)( 7,186)
( 8,190)( 9,189)( 10,188)( 11,192)( 12,191)( 13,195)( 14,194)( 15,193)
( 16,222)( 17,221)( 18,225)( 19,224)( 20,223)( 21,212)( 22,211)( 23,215)
( 24,214)( 25,213)( 26,217)( 27,216)( 28,220)( 29,219)( 30,218)( 31,202)
( 32,201)( 33,205)( 34,204)( 35,203)( 36,207)( 37,206)( 38,210)( 39,209)
( 40,208)( 41,197)( 42,196)( 43,200)( 44,199)( 45,198)( 46,227)( 47,226)
( 48,230)( 49,229)( 50,228)( 51,232)( 52,231)( 53,235)( 54,234)( 55,233)
( 56,237)( 57,236)( 58,240)( 59,239)( 60,238)( 61,267)( 62,266)( 63,270)
( 64,269)( 65,268)( 66,257)( 67,256)( 68,260)( 69,259)( 70,258)( 71,262)
( 72,261)( 73,265)( 74,264)( 75,263)( 76,247)( 77,246)( 78,250)( 79,249)
( 80,248)( 81,252)( 82,251)( 83,255)( 84,254)( 85,253)( 86,242)( 87,241)
( 88,245)( 89,244)( 90,243)( 91,272)( 92,271)( 93,275)( 94,274)( 95,273)
( 96,277)( 97,276)( 98,280)( 99,279)(100,278)(101,282)(102,281)(103,285)
(104,284)(105,283)(106,312)(107,311)(108,315)(109,314)(110,313)(111,302)
(112,301)(113,305)(114,304)(115,303)(116,307)(117,306)(118,310)(119,309)
(120,308)(121,292)(122,291)(123,295)(124,294)(125,293)(126,297)(127,296)
(128,300)(129,299)(130,298)(131,287)(132,286)(133,290)(134,289)(135,288)
(136,317)(137,316)(138,320)(139,319)(140,318)(141,322)(142,321)(143,325)
(144,324)(145,323)(146,327)(147,326)(148,330)(149,329)(150,328)(151,357)
(152,356)(153,360)(154,359)(155,358)(156,347)(157,346)(158,350)(159,349)
(160,348)(161,352)(162,351)(163,355)(164,354)(165,353)(166,337)(167,336)
(168,340)(169,339)(170,338)(171,342)(172,341)(173,345)(174,344)(175,343)
(176,332)(177,331)(178,335)(179,334)(180,333);
s2 := Sym(360)!( 1,111)( 2,112)( 3,113)( 4,114)( 5,115)( 6,121)( 7,122)
( 8,123)( 9,124)( 10,125)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,126)( 17,127)( 18,128)( 19,129)( 20,130)( 21, 91)( 22, 92)( 23, 93)
( 24, 94)( 25, 95)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31, 96)
( 32, 97)( 33, 98)( 34, 99)( 35,100)( 36,106)( 37,107)( 38,108)( 39,109)
( 40,110)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,156)( 47,157)
( 48,158)( 49,159)( 50,160)( 51,166)( 52,167)( 53,168)( 54,169)( 55,170)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)
( 64,174)( 65,175)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,161)
( 72,162)( 73,163)( 74,164)( 75,165)( 76,141)( 77,142)( 78,143)( 79,144)
( 80,145)( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180)(181,201)(182,202)(183,203)(184,204)(185,205)
(186,211)(187,212)(188,213)(189,214)(190,215)(196,216)(197,217)(198,218)
(199,219)(200,220)(226,246)(227,247)(228,248)(229,249)(230,250)(231,256)
(232,257)(233,258)(234,259)(235,260)(241,261)(242,262)(243,263)(244,264)
(245,265)(271,291)(272,292)(273,293)(274,294)(275,295)(276,301)(277,302)
(278,303)(279,304)(280,305)(286,306)(287,307)(288,308)(289,309)(290,310)
(316,336)(317,337)(318,338)(319,339)(320,340)(321,346)(322,347)(323,348)
(324,349)(325,350)(331,351)(332,352)(333,353)(334,354)(335,355);
poly := sub<Sym(360)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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 >;
References : None.
to this polytope