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Polytope of Type {16,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,20}*640a
Also Known As : {16,20|2}. if this polytope has another name.
Group : SmallGroup(640,2066)
Rank : 3
Schlafli Type : {16,20}
Number of vertices, edges, etc : 16, 160, 20
Order of s0s1s2 : 80
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {16,20,2} of size 1280
Vertex Figure Of :
   {2,16,20} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,20}*320a, {16,10}*320
   4-fold quotients : {4,20}*160, {8,10}*160
   5-fold quotients : {16,4}*128a
   8-fold quotients : {2,20}*80, {4,10}*80
   10-fold quotients : {8,4}*64a, {16,2}*64
   16-fold quotients : {2,10}*40
   20-fold quotients : {4,4}*32, {8,2}*32
   32-fold quotients : {2,5}*20
   40-fold quotients : {2,4}*16, {4,2}*16
   80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {16,20}*1280a, {16,40}*1280c, {16,40}*1280d, {32,20}*1280a, {32,20}*1280b
   3-fold covers : {16,60}*1920a, {48,20}*1920a
Permutation Representation (GAP) :
s0 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)
(  9,169)( 10,170)( 11,176)( 12,177)( 13,178)( 14,179)( 15,180)( 16,171)
( 17,172)( 18,173)( 19,174)( 20,175)( 21,181)( 22,182)( 23,183)( 24,184)
( 25,185)( 26,186)( 27,187)( 28,188)( 29,189)( 30,190)( 31,196)( 32,197)
( 33,198)( 34,199)( 35,200)( 36,191)( 37,192)( 38,193)( 39,194)( 40,195)
( 41,211)( 42,212)( 43,213)( 44,214)( 45,215)( 46,216)( 47,217)( 48,218)
( 49,219)( 50,220)( 51,201)( 52,202)( 53,203)( 54,204)( 55,205)( 56,206)
( 57,207)( 58,208)( 59,209)( 60,210)( 61,231)( 62,232)( 63,233)( 64,234)
( 65,235)( 66,236)( 67,237)( 68,238)( 69,239)( 70,240)( 71,221)( 72,222)
( 73,223)( 74,224)( 75,225)( 76,226)( 77,227)( 78,228)( 79,229)( 80,230)
( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)( 88,248)
( 89,249)( 90,250)( 91,256)( 92,257)( 93,258)( 94,259)( 95,260)( 96,251)
( 97,252)( 98,253)( 99,254)(100,255)(101,261)(102,262)(103,263)(104,264)
(105,265)(106,266)(107,267)(108,268)(109,269)(110,270)(111,276)(112,277)
(113,278)(114,279)(115,280)(116,271)(117,272)(118,273)(119,274)(120,275)
(121,291)(122,292)(123,293)(124,294)(125,295)(126,296)(127,297)(128,298)
(129,299)(130,300)(131,281)(132,282)(133,283)(134,284)(135,285)(136,286)
(137,287)(138,288)(139,289)(140,290)(141,311)(142,312)(143,313)(144,314)
(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(151,301)(152,302)
(153,303)(154,304)(155,305)(156,306)(157,307)(158,308)(159,309)(160,310);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 11, 16)( 12, 20)( 13, 19)( 14, 18)
( 15, 17)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 31, 36)( 32, 40)( 33, 39)
( 34, 38)( 35, 37)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 46, 56)
( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 71)( 62, 75)( 63, 74)( 64, 73)
( 65, 72)( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,101)( 82,105)
( 83,104)( 84,103)( 85,102)( 86,106)( 87,110)( 88,109)( 89,108)( 90,107)
( 91,116)( 92,120)( 93,119)( 94,118)( 95,117)( 96,111)( 97,115)( 98,114)
( 99,113)(100,112)(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)
(127,160)(128,159)(129,158)(130,157)(131,141)(132,145)(133,144)(134,143)
(135,142)(136,146)(137,150)(138,149)(139,148)(140,147)(161,201)(162,205)
(163,204)(164,203)(165,202)(166,206)(167,210)(168,209)(169,208)(170,207)
(171,216)(172,220)(173,219)(174,218)(175,217)(176,211)(177,215)(178,214)
(179,213)(180,212)(181,221)(182,225)(183,224)(184,223)(185,222)(186,226)
(187,230)(188,229)(189,228)(190,227)(191,236)(192,240)(193,239)(194,238)
(195,237)(196,231)(197,235)(198,234)(199,233)(200,232)(241,301)(242,305)
(243,304)(244,303)(245,302)(246,306)(247,310)(248,309)(249,308)(250,307)
(251,316)(252,320)(253,319)(254,318)(255,317)(256,311)(257,315)(258,314)
(259,313)(260,312)(261,281)(262,285)(263,284)(264,283)(265,282)(266,286)
(267,290)(268,289)(269,288)(270,287)(271,296)(272,300)(273,299)(274,298)
(275,297)(276,291)(277,295)(278,294)(279,293)(280,292);;
s2 := (  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 87)(  7, 86)(  8, 90)
(  9, 89)( 10, 88)( 11, 92)( 12, 91)( 13, 95)( 14, 94)( 15, 93)( 16, 97)
( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21,102)( 22,101)( 23,105)( 24,104)
( 25,103)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,112)( 32,111)
( 33,115)( 34,114)( 35,113)( 36,117)( 37,116)( 38,120)( 39,119)( 40,118)
( 41,122)( 42,121)( 43,125)( 44,124)( 45,123)( 46,127)( 47,126)( 48,130)
( 49,129)( 50,128)( 51,132)( 52,131)( 53,135)( 54,134)( 55,133)( 56,137)
( 57,136)( 58,140)( 59,139)( 60,138)( 61,142)( 62,141)( 63,145)( 64,144)
( 65,143)( 66,147)( 67,146)( 68,150)( 69,149)( 70,148)( 71,152)( 72,151)
( 73,155)( 74,154)( 75,153)( 76,157)( 77,156)( 78,160)( 79,159)( 80,158)
(161,242)(162,241)(163,245)(164,244)(165,243)(166,247)(167,246)(168,250)
(169,249)(170,248)(171,252)(172,251)(173,255)(174,254)(175,253)(176,257)
(177,256)(178,260)(179,259)(180,258)(181,262)(182,261)(183,265)(184,264)
(185,263)(186,267)(187,266)(188,270)(189,269)(190,268)(191,272)(192,271)
(193,275)(194,274)(195,273)(196,277)(197,276)(198,280)(199,279)(200,278)
(201,282)(202,281)(203,285)(204,284)(205,283)(206,287)(207,286)(208,290)
(209,289)(210,288)(211,292)(212,291)(213,295)(214,294)(215,293)(216,297)
(217,296)(218,300)(219,299)(220,298)(221,302)(222,301)(223,305)(224,304)
(225,303)(226,307)(227,306)(228,310)(229,309)(230,308)(231,312)(232,311)
(233,315)(234,314)(235,313)(236,317)(237,316)(238,320)(239,319)(240,318);;
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*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(320)!(  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)
(  8,168)(  9,169)( 10,170)( 11,176)( 12,177)( 13,178)( 14,179)( 15,180)
( 16,171)( 17,172)( 18,173)( 19,174)( 20,175)( 21,181)( 22,182)( 23,183)
( 24,184)( 25,185)( 26,186)( 27,187)( 28,188)( 29,189)( 30,190)( 31,196)
( 32,197)( 33,198)( 34,199)( 35,200)( 36,191)( 37,192)( 38,193)( 39,194)
( 40,195)( 41,211)( 42,212)( 43,213)( 44,214)( 45,215)( 46,216)( 47,217)
( 48,218)( 49,219)( 50,220)( 51,201)( 52,202)( 53,203)( 54,204)( 55,205)
( 56,206)( 57,207)( 58,208)( 59,209)( 60,210)( 61,231)( 62,232)( 63,233)
( 64,234)( 65,235)( 66,236)( 67,237)( 68,238)( 69,239)( 70,240)( 71,221)
( 72,222)( 73,223)( 74,224)( 75,225)( 76,226)( 77,227)( 78,228)( 79,229)
( 80,230)( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)
( 88,248)( 89,249)( 90,250)( 91,256)( 92,257)( 93,258)( 94,259)( 95,260)
( 96,251)( 97,252)( 98,253)( 99,254)(100,255)(101,261)(102,262)(103,263)
(104,264)(105,265)(106,266)(107,267)(108,268)(109,269)(110,270)(111,276)
(112,277)(113,278)(114,279)(115,280)(116,271)(117,272)(118,273)(119,274)
(120,275)(121,291)(122,292)(123,293)(124,294)(125,295)(126,296)(127,297)
(128,298)(129,299)(130,300)(131,281)(132,282)(133,283)(134,284)(135,285)
(136,286)(137,287)(138,288)(139,289)(140,290)(141,311)(142,312)(143,313)
(144,314)(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(151,301)
(152,302)(153,303)(154,304)(155,305)(156,306)(157,307)(158,308)(159,309)
(160,310);
s1 := Sym(320)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 11, 16)( 12, 20)( 13, 19)
( 14, 18)( 15, 17)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)
( 46, 56)( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 71)( 62, 75)( 63, 74)
( 64, 73)( 65, 72)( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,101)
( 82,105)( 83,104)( 84,103)( 85,102)( 86,106)( 87,110)( 88,109)( 89,108)
( 90,107)( 91,116)( 92,120)( 93,119)( 94,118)( 95,117)( 96,111)( 97,115)
( 98,114)( 99,113)(100,112)(121,151)(122,155)(123,154)(124,153)(125,152)
(126,156)(127,160)(128,159)(129,158)(130,157)(131,141)(132,145)(133,144)
(134,143)(135,142)(136,146)(137,150)(138,149)(139,148)(140,147)(161,201)
(162,205)(163,204)(164,203)(165,202)(166,206)(167,210)(168,209)(169,208)
(170,207)(171,216)(172,220)(173,219)(174,218)(175,217)(176,211)(177,215)
(178,214)(179,213)(180,212)(181,221)(182,225)(183,224)(184,223)(185,222)
(186,226)(187,230)(188,229)(189,228)(190,227)(191,236)(192,240)(193,239)
(194,238)(195,237)(196,231)(197,235)(198,234)(199,233)(200,232)(241,301)
(242,305)(243,304)(244,303)(245,302)(246,306)(247,310)(248,309)(249,308)
(250,307)(251,316)(252,320)(253,319)(254,318)(255,317)(256,311)(257,315)
(258,314)(259,313)(260,312)(261,281)(262,285)(263,284)(264,283)(265,282)
(266,286)(267,290)(268,289)(269,288)(270,287)(271,296)(272,300)(273,299)
(274,298)(275,297)(276,291)(277,295)(278,294)(279,293)(280,292);
s2 := Sym(320)!(  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 87)(  7, 86)
(  8, 90)(  9, 89)( 10, 88)( 11, 92)( 12, 91)( 13, 95)( 14, 94)( 15, 93)
( 16, 97)( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21,102)( 22,101)( 23,105)
( 24,104)( 25,103)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,112)
( 32,111)( 33,115)( 34,114)( 35,113)( 36,117)( 37,116)( 38,120)( 39,119)
( 40,118)( 41,122)( 42,121)( 43,125)( 44,124)( 45,123)( 46,127)( 47,126)
( 48,130)( 49,129)( 50,128)( 51,132)( 52,131)( 53,135)( 54,134)( 55,133)
( 56,137)( 57,136)( 58,140)( 59,139)( 60,138)( 61,142)( 62,141)( 63,145)
( 64,144)( 65,143)( 66,147)( 67,146)( 68,150)( 69,149)( 70,148)( 71,152)
( 72,151)( 73,155)( 74,154)( 75,153)( 76,157)( 77,156)( 78,160)( 79,159)
( 80,158)(161,242)(162,241)(163,245)(164,244)(165,243)(166,247)(167,246)
(168,250)(169,249)(170,248)(171,252)(172,251)(173,255)(174,254)(175,253)
(176,257)(177,256)(178,260)(179,259)(180,258)(181,262)(182,261)(183,265)
(184,264)(185,263)(186,267)(187,266)(188,270)(189,269)(190,268)(191,272)
(192,271)(193,275)(194,274)(195,273)(196,277)(197,276)(198,280)(199,279)
(200,278)(201,282)(202,281)(203,285)(204,284)(205,283)(206,287)(207,286)
(208,290)(209,289)(210,288)(211,292)(212,291)(213,295)(214,294)(215,293)
(216,297)(217,296)(218,300)(219,299)(220,298)(221,302)(222,301)(223,305)
(224,304)(225,303)(226,307)(227,306)(228,310)(229,309)(230,308)(231,312)
(232,311)(233,315)(234,314)(235,313)(236,317)(237,316)(238,320)(239,319)
(240,318);
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*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope