Polytope of Type {20,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4}*640b
if this polytope has a name.
Group : SmallGroup(640,21459)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 80, 160, 16
Order of s0s1s2 : 20
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{20,4,2} of size 1280
Vertex Figure Of :
{2,20,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,4}*320a
4-fold quotients : {5,4}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,8}*1280i, {20,8}*1280j, {40,4}*1280e, {40,4}*1280f, {20,4}*1280c
3-fold covers : {60,4}*1920f
Permutation Representation (GAP) :
s0 := ( 2, 9)( 3, 16)( 4, 8)( 6, 13)( 7, 12)( 11, 15)( 17, 65)( 18, 73)
( 19, 80)( 20, 72)( 21, 69)( 22, 77)( 23, 76)( 24, 68)( 25, 66)( 26, 74)
( 27, 79)( 28, 71)( 29, 70)( 30, 78)( 31, 75)( 32, 67)( 33, 49)( 34, 57)
( 35, 64)( 36, 56)( 37, 53)( 38, 61)( 39, 60)( 40, 52)( 41, 50)( 42, 58)
( 43, 63)( 44, 55)( 45, 54)( 46, 62)( 47, 59)( 48, 51)( 82, 89)( 83, 96)
( 84, 88)( 86, 93)( 87, 92)( 91, 95)( 97,145)( 98,153)( 99,160)(100,152)
(101,149)(102,157)(103,156)(104,148)(105,146)(106,154)(107,159)(108,151)
(109,150)(110,158)(111,155)(112,147)(113,129)(114,137)(115,144)(116,136)
(117,133)(118,141)(119,140)(120,132)(121,130)(122,138)(123,143)(124,135)
(125,134)(126,142)(127,139)(128,131)(161,241)(162,249)(163,256)(164,248)
(165,245)(166,253)(167,252)(168,244)(169,242)(170,250)(171,255)(172,247)
(173,246)(174,254)(175,251)(176,243)(177,305)(178,313)(179,320)(180,312)
(181,309)(182,317)(183,316)(184,308)(185,306)(186,314)(187,319)(188,311)
(189,310)(190,318)(191,315)(192,307)(193,289)(194,297)(195,304)(196,296)
(197,293)(198,301)(199,300)(200,292)(201,290)(202,298)(203,303)(204,295)
(205,294)(206,302)(207,299)(208,291)(209,273)(210,281)(211,288)(212,280)
(213,277)(214,285)(215,284)(216,276)(217,274)(218,282)(219,287)(220,279)
(221,278)(222,286)(223,283)(224,275)(225,257)(226,265)(227,272)(228,264)
(229,261)(230,269)(231,268)(232,260)(233,258)(234,266)(235,271)(236,263)
(237,262)(238,270)(239,267)(240,259);;
s1 := ( 1,177)( 2,187)( 3,186)( 4,180)( 5,189)( 6,183)( 7,182)( 8,192)
( 9,185)( 10,179)( 11,178)( 12,188)( 13,181)( 14,191)( 15,190)( 16,184)
( 17,161)( 18,171)( 19,170)( 20,164)( 21,173)( 22,167)( 23,166)( 24,176)
( 25,169)( 26,163)( 27,162)( 28,172)( 29,165)( 30,175)( 31,174)( 32,168)
( 33,225)( 34,235)( 35,234)( 36,228)( 37,237)( 38,231)( 39,230)( 40,240)
( 41,233)( 42,227)( 43,226)( 44,236)( 45,229)( 46,239)( 47,238)( 48,232)
( 49,209)( 50,219)( 51,218)( 52,212)( 53,221)( 54,215)( 55,214)( 56,224)
( 57,217)( 58,211)( 59,210)( 60,220)( 61,213)( 62,223)( 63,222)( 64,216)
( 65,193)( 66,203)( 67,202)( 68,196)( 69,205)( 70,199)( 71,198)( 72,208)
( 73,201)( 74,195)( 75,194)( 76,204)( 77,197)( 78,207)( 79,206)( 80,200)
( 81,257)( 82,267)( 83,266)( 84,260)( 85,269)( 86,263)( 87,262)( 88,272)
( 89,265)( 90,259)( 91,258)( 92,268)( 93,261)( 94,271)( 95,270)( 96,264)
( 97,241)( 98,251)( 99,250)(100,244)(101,253)(102,247)(103,246)(104,256)
(105,249)(106,243)(107,242)(108,252)(109,245)(110,255)(111,254)(112,248)
(113,305)(114,315)(115,314)(116,308)(117,317)(118,311)(119,310)(120,320)
(121,313)(122,307)(123,306)(124,316)(125,309)(126,319)(127,318)(128,312)
(129,289)(130,299)(131,298)(132,292)(133,301)(134,295)(135,294)(136,304)
(137,297)(138,291)(139,290)(140,300)(141,293)(142,303)(143,302)(144,296)
(145,273)(146,283)(147,282)(148,276)(149,285)(150,279)(151,278)(152,288)
(153,281)(154,275)(155,274)(156,284)(157,277)(158,287)(159,286)(160,280);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)(108,112)
(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)
(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)
(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)
(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)
(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)
(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)
(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224)
(225,229)(226,230)(227,231)(228,232)(233,237)(234,238)(235,239)(236,240)
(241,245)(242,246)(243,247)(244,248)(249,253)(250,254)(251,255)(252,256)
(257,261)(258,262)(259,263)(260,264)(265,269)(266,270)(267,271)(268,272)
(273,277)(274,278)(275,279)(276,280)(281,285)(282,286)(283,287)(284,288)
(289,293)(290,294)(291,295)(292,296)(297,301)(298,302)(299,303)(300,304)
(305,309)(306,310)(307,311)(308,312)(313,317)(314,318)(315,319)(316,320);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*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(320)!( 2, 9)( 3, 16)( 4, 8)( 6, 13)( 7, 12)( 11, 15)( 17, 65)
( 18, 73)( 19, 80)( 20, 72)( 21, 69)( 22, 77)( 23, 76)( 24, 68)( 25, 66)
( 26, 74)( 27, 79)( 28, 71)( 29, 70)( 30, 78)( 31, 75)( 32, 67)( 33, 49)
( 34, 57)( 35, 64)( 36, 56)( 37, 53)( 38, 61)( 39, 60)( 40, 52)( 41, 50)
( 42, 58)( 43, 63)( 44, 55)( 45, 54)( 46, 62)( 47, 59)( 48, 51)( 82, 89)
( 83, 96)( 84, 88)( 86, 93)( 87, 92)( 91, 95)( 97,145)( 98,153)( 99,160)
(100,152)(101,149)(102,157)(103,156)(104,148)(105,146)(106,154)(107,159)
(108,151)(109,150)(110,158)(111,155)(112,147)(113,129)(114,137)(115,144)
(116,136)(117,133)(118,141)(119,140)(120,132)(121,130)(122,138)(123,143)
(124,135)(125,134)(126,142)(127,139)(128,131)(161,241)(162,249)(163,256)
(164,248)(165,245)(166,253)(167,252)(168,244)(169,242)(170,250)(171,255)
(172,247)(173,246)(174,254)(175,251)(176,243)(177,305)(178,313)(179,320)
(180,312)(181,309)(182,317)(183,316)(184,308)(185,306)(186,314)(187,319)
(188,311)(189,310)(190,318)(191,315)(192,307)(193,289)(194,297)(195,304)
(196,296)(197,293)(198,301)(199,300)(200,292)(201,290)(202,298)(203,303)
(204,295)(205,294)(206,302)(207,299)(208,291)(209,273)(210,281)(211,288)
(212,280)(213,277)(214,285)(215,284)(216,276)(217,274)(218,282)(219,287)
(220,279)(221,278)(222,286)(223,283)(224,275)(225,257)(226,265)(227,272)
(228,264)(229,261)(230,269)(231,268)(232,260)(233,258)(234,266)(235,271)
(236,263)(237,262)(238,270)(239,267)(240,259);
s1 := Sym(320)!( 1,177)( 2,187)( 3,186)( 4,180)( 5,189)( 6,183)( 7,182)
( 8,192)( 9,185)( 10,179)( 11,178)( 12,188)( 13,181)( 14,191)( 15,190)
( 16,184)( 17,161)( 18,171)( 19,170)( 20,164)( 21,173)( 22,167)( 23,166)
( 24,176)( 25,169)( 26,163)( 27,162)( 28,172)( 29,165)( 30,175)( 31,174)
( 32,168)( 33,225)( 34,235)( 35,234)( 36,228)( 37,237)( 38,231)( 39,230)
( 40,240)( 41,233)( 42,227)( 43,226)( 44,236)( 45,229)( 46,239)( 47,238)
( 48,232)( 49,209)( 50,219)( 51,218)( 52,212)( 53,221)( 54,215)( 55,214)
( 56,224)( 57,217)( 58,211)( 59,210)( 60,220)( 61,213)( 62,223)( 63,222)
( 64,216)( 65,193)( 66,203)( 67,202)( 68,196)( 69,205)( 70,199)( 71,198)
( 72,208)( 73,201)( 74,195)( 75,194)( 76,204)( 77,197)( 78,207)( 79,206)
( 80,200)( 81,257)( 82,267)( 83,266)( 84,260)( 85,269)( 86,263)( 87,262)
( 88,272)( 89,265)( 90,259)( 91,258)( 92,268)( 93,261)( 94,271)( 95,270)
( 96,264)( 97,241)( 98,251)( 99,250)(100,244)(101,253)(102,247)(103,246)
(104,256)(105,249)(106,243)(107,242)(108,252)(109,245)(110,255)(111,254)
(112,248)(113,305)(114,315)(115,314)(116,308)(117,317)(118,311)(119,310)
(120,320)(121,313)(122,307)(123,306)(124,316)(125,309)(126,319)(127,318)
(128,312)(129,289)(130,299)(131,298)(132,292)(133,301)(134,295)(135,294)
(136,304)(137,297)(138,291)(139,290)(140,300)(141,293)(142,303)(143,302)
(144,296)(145,273)(146,283)(147,282)(148,276)(149,285)(150,279)(151,278)
(152,288)(153,281)(154,275)(155,274)(156,284)(157,277)(158,287)(159,286)
(160,280);
s2 := Sym(320)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)
(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)
(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)
(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)
(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)
(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)
(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)
(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)
(220,224)(225,229)(226,230)(227,231)(228,232)(233,237)(234,238)(235,239)
(236,240)(241,245)(242,246)(243,247)(244,248)(249,253)(250,254)(251,255)
(252,256)(257,261)(258,262)(259,263)(260,264)(265,269)(266,270)(267,271)
(268,272)(273,277)(274,278)(275,279)(276,280)(281,285)(282,286)(283,287)
(284,288)(289,293)(290,294)(291,295)(292,296)(297,301)(298,302)(299,303)
(300,304)(305,309)(306,310)(307,311)(308,312)(313,317)(314,318)(315,319)
(316,320);
poly := sub<Sym(320)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*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