Polytope of Type {4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*640c
if this polytope has a name.
Group : SmallGroup(640,21459)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 16, 160, 80
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,20,2} of size 1280
Vertex Figure Of :
   {2,4,20} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,10}*320a
   4-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,20}*1280k, {8,20}*1280l, {4,40}*1280g, {4,40}*1280h, {4,20}*1280c
   3-fold covers : {4,60}*1920g
Permutation Representation (GAP) :

s0 := (  1, 89)(  2, 90)(  3, 91)(  4, 92)(  5, 93)(  6, 94)(  7, 95)(  8, 96)
(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17,105)( 18,106)( 19,107)( 20,108)( 21,109)( 22,110)( 23,111)( 24,112)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,121)( 34,122)( 35,123)( 36,124)( 37,125)( 38,126)( 39,127)( 40,128)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,137)( 50,138)( 51,139)( 52,140)( 53,141)( 54,142)( 55,143)( 56,144)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,153)( 66,154)( 67,155)( 68,156)( 69,157)( 70,158)( 71,159)( 72,160)
( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)( 80,152)
(161,249)(162,250)(163,251)(164,252)(165,253)(166,254)(167,255)(168,256)
(169,241)(170,242)(171,243)(172,244)(173,245)(174,246)(175,247)(176,248)
(177,265)(178,266)(179,267)(180,268)(181,269)(182,270)(183,271)(184,272)
(185,257)(186,258)(187,259)(188,260)(189,261)(190,262)(191,263)(192,264)
(193,281)(194,282)(195,283)(196,284)(197,285)(198,286)(199,287)(200,288)
(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)
(209,297)(210,298)(211,299)(212,300)(213,301)(214,302)(215,303)(216,304)
(217,289)(218,290)(219,291)(220,292)(221,293)(222,294)(223,295)(224,296)
(225,313)(226,314)(227,315)(228,316)(229,317)(230,318)(231,319)(232,320)
(233,305)(234,306)(235,307)(236,308)(237,309)(238,310)(239,311)(240,312);;
s1 := (  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);;
s2 := (  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);;
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*s0*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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope