Polytope of Type {4,4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,20}*640
Also Known As : {{4,4|2},{4,20|2}}. if this polytope has another name.
Group : SmallGroup(640,8697)
Rank : 4
Schlafli Type : {4,4,20}
Number of vertices, edges, etc : 4, 8, 40, 20
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,20,2} of size 1280
Vertex Figure Of :
   {2,4,4,20} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,20}*320, {4,2,20}*320, {4,4,10}*320
   4-fold quotients : {2,2,20}*160, {2,4,10}*160, {4,2,10}*160
   5-fold quotients : {4,4,4}*128
   8-fold quotients : {4,2,5}*80, {2,2,10}*80
   10-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   16-fold quotients : {2,2,5}*40
   20-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
   40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,4,20}*1280a, {4,4,40}*1280a, {8,4,20}*1280b, {4,4,40}*1280b, {4,8,20}*1280a, {4,4,20}*1280a, {4,4,20}*1280b, {4,8,20}*1280b, {4,8,20}*1280c, {4,8,20}*1280d
   3-fold covers : {4,4,60}*1920, {4,12,20}*1920a, {12,4,20}*1920
Permutation Representation (GAP) : 
s0 := (  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 56)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 61)( 22, 62)( 23, 63)( 24, 64)
( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 69)( 30, 70)( 31, 71)( 32, 72)
( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)( 40, 80)
( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)( 88,128)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)( 96,136)
( 97,137)( 98,138)( 99,139)(100,140)(101,141)(102,142)(103,143)(104,144)
(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)
(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)
(161,201)(162,202)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)
(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)
(177,217)(178,218)(179,219)(180,220)(181,221)(182,222)(183,223)(184,224)
(185,225)(186,226)(187,227)(188,228)(189,229)(190,230)(191,231)(192,232)
(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)
(241,281)(242,282)(243,283)(244,284)(245,285)(246,286)(247,287)(248,288)
(249,289)(250,290)(251,291)(252,292)(253,293)(254,294)(255,295)(256,296)
(257,297)(258,298)(259,299)(260,300)(261,301)(262,302)(263,303)(264,304)
(265,305)(266,306)(267,307)(268,308)(269,309)(270,310)(271,311)(272,312)
(273,313)(274,314)(275,315)(276,316)(277,317)(278,318)(279,319)(280,320);;
s1 := ( 41, 61)( 42, 62)( 43, 63)( 44, 64)( 45, 65)( 46, 66)( 47, 67)( 48, 68)
( 49, 69)( 50, 70)( 51, 71)( 52, 72)( 53, 73)( 54, 74)( 55, 75)( 56, 76)
( 57, 77)( 58, 78)( 59, 79)( 60, 80)( 81, 91)( 82, 92)( 83, 93)( 84, 94)
( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,111)(102,112)
(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)
(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)
(129,159)(130,160)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)
(137,147)(138,148)(139,149)(140,150)(161,181)(162,182)(163,183)(164,184)
(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)
(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)
(241,271)(242,272)(243,273)(244,274)(245,275)(246,276)(247,277)(248,278)
(249,279)(250,280)(251,261)(252,262)(253,263)(254,264)(255,265)(256,266)
(257,267)(258,268)(259,269)(260,270)(281,291)(282,292)(283,293)(284,294)
(285,295)(286,296)(287,297)(288,298)(289,299)(290,300)(301,311)(302,312)
(303,313)(304,314)(305,315)(306,316)(307,317)(308,318)(309,319)(310,320);;
s2 := (  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)(  8, 89)
(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)( 16, 96)
( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,101)( 22,105)( 23,104)( 24,103)
( 25,102)( 26,106)( 27,110)( 28,109)( 29,108)( 30,107)( 31,111)( 32,115)
( 33,114)( 34,113)( 35,112)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)
( 41,121)( 42,125)( 43,124)( 44,123)( 45,122)( 46,126)( 47,130)( 48,129)
( 49,128)( 50,127)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)( 56,136)
( 57,140)( 58,139)( 59,138)( 60,137)( 61,141)( 62,145)( 63,144)( 64,143)
( 65,142)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,151)( 72,155)
( 73,154)( 74,153)( 75,152)( 76,156)( 77,160)( 78,159)( 79,158)( 80,157)
(161,241)(162,245)(163,244)(164,243)(165,242)(166,246)(167,250)(168,249)
(169,248)(170,247)(171,251)(172,255)(173,254)(174,253)(175,252)(176,256)
(177,260)(178,259)(179,258)(180,257)(181,261)(182,265)(183,264)(184,263)
(185,262)(186,266)(187,270)(188,269)(189,268)(190,267)(191,271)(192,275)
(193,274)(194,273)(195,272)(196,276)(197,280)(198,279)(199,278)(200,277)
(201,281)(202,285)(203,284)(204,283)(205,282)(206,286)(207,290)(208,289)
(209,288)(210,287)(211,291)(212,295)(213,294)(214,293)(215,292)(216,296)
(217,300)(218,299)(219,298)(220,297)(221,301)(222,305)(223,304)(224,303)
(225,302)(226,306)(227,310)(228,309)(229,308)(230,307)(231,311)(232,315)
(233,314)(234,313)(235,312)(236,316)(237,320)(238,319)(239,318)(240,317);;
s3 := (  1,202)(  2,201)(  3,205)(  4,204)(  5,203)(  6,207)(  7,206)(  8,210)
(  9,209)( 10,208)( 11,212)( 12,211)( 13,215)( 14,214)( 15,213)( 16,217)
( 17,216)( 18,220)( 19,219)( 20,218)( 21,222)( 22,221)( 23,225)( 24,224)
( 25,223)( 26,227)( 27,226)( 28,230)( 29,229)( 30,228)( 31,232)( 32,231)
( 33,235)( 34,234)( 35,233)( 36,237)( 37,236)( 38,240)( 39,239)( 40,238)
( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)( 48,170)
( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)( 56,177)
( 57,176)( 58,180)( 59,179)( 60,178)( 61,182)( 62,181)( 63,185)( 64,184)
( 65,183)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,192)( 72,191)
( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)( 80,198)
( 81,287)( 82,286)( 83,290)( 84,289)( 85,288)( 86,282)( 87,281)( 88,285)
( 89,284)( 90,283)( 91,297)( 92,296)( 93,300)( 94,299)( 95,298)( 96,292)
( 97,291)( 98,295)( 99,294)(100,293)(101,307)(102,306)(103,310)(104,309)
(105,308)(106,302)(107,301)(108,305)(109,304)(110,303)(111,317)(112,316)
(113,320)(114,319)(115,318)(116,312)(117,311)(118,315)(119,314)(120,313)
(121,247)(122,246)(123,250)(124,249)(125,248)(126,242)(127,241)(128,245)
(129,244)(130,243)(131,257)(132,256)(133,260)(134,259)(135,258)(136,252)
(137,251)(138,255)(139,254)(140,253)(141,267)(142,266)(143,270)(144,269)
(145,268)(146,262)(147,261)(148,265)(149,264)(150,263)(151,277)(152,276)
(153,280)(154,279)(155,278)(156,272)(157,271)(158,275)(159,274)(160,273);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(320)!(  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)
( 16, 56)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 61)( 22, 62)( 23, 63)
( 24, 64)( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 69)( 30, 70)( 31, 71)
( 32, 72)( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)
( 40, 80)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)
( 88,128)( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)
( 96,136)( 97,137)( 98,138)( 99,139)(100,140)(101,141)(102,142)(103,143)
(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)
(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)
(120,160)(161,201)(162,202)(163,203)(164,204)(165,205)(166,206)(167,207)
(168,208)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)
(176,216)(177,217)(178,218)(179,219)(180,220)(181,221)(182,222)(183,223)
(184,224)(185,225)(186,226)(187,227)(188,228)(189,229)(190,230)(191,231)
(192,232)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)
(200,240)(241,281)(242,282)(243,283)(244,284)(245,285)(246,286)(247,287)
(248,288)(249,289)(250,290)(251,291)(252,292)(253,293)(254,294)(255,295)
(256,296)(257,297)(258,298)(259,299)(260,300)(261,301)(262,302)(263,303)
(264,304)(265,305)(266,306)(267,307)(268,308)(269,309)(270,310)(271,311)
(272,312)(273,313)(274,314)(275,315)(276,316)(277,317)(278,318)(279,319)
(280,320);
s1 := Sym(320)!( 41, 61)( 42, 62)( 43, 63)( 44, 64)( 45, 65)( 46, 66)( 47, 67)
( 48, 68)( 49, 69)( 50, 70)( 51, 71)( 52, 72)( 53, 73)( 54, 74)( 55, 75)
( 56, 76)( 57, 77)( 58, 78)( 59, 79)( 60, 80)( 81, 91)( 82, 92)( 83, 93)
( 84, 94)( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,111)
(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)
(110,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)
(128,158)(129,159)(130,160)(131,141)(132,142)(133,143)(134,144)(135,145)
(136,146)(137,147)(138,148)(139,149)(140,150)(161,181)(162,182)(163,183)
(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)
(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)
(180,200)(241,271)(242,272)(243,273)(244,274)(245,275)(246,276)(247,277)
(248,278)(249,279)(250,280)(251,261)(252,262)(253,263)(254,264)(255,265)
(256,266)(257,267)(258,268)(259,269)(260,270)(281,291)(282,292)(283,293)
(284,294)(285,295)(286,296)(287,297)(288,298)(289,299)(290,300)(301,311)
(302,312)(303,313)(304,314)(305,315)(306,316)(307,317)(308,318)(309,319)
(310,320);
s2 := Sym(320)!(  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)
(  8, 89)(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)
( 16, 96)( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,101)( 22,105)( 23,104)
( 24,103)( 25,102)( 26,106)( 27,110)( 28,109)( 29,108)( 30,107)( 31,111)
( 32,115)( 33,114)( 34,113)( 35,112)( 36,116)( 37,120)( 38,119)( 39,118)
( 40,117)( 41,121)( 42,125)( 43,124)( 44,123)( 45,122)( 46,126)( 47,130)
( 48,129)( 49,128)( 50,127)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)
( 56,136)( 57,140)( 58,139)( 59,138)( 60,137)( 61,141)( 62,145)( 63,144)
( 64,143)( 65,142)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,151)
( 72,155)( 73,154)( 74,153)( 75,152)( 76,156)( 77,160)( 78,159)( 79,158)
( 80,157)(161,241)(162,245)(163,244)(164,243)(165,242)(166,246)(167,250)
(168,249)(169,248)(170,247)(171,251)(172,255)(173,254)(174,253)(175,252)
(176,256)(177,260)(178,259)(179,258)(180,257)(181,261)(182,265)(183,264)
(184,263)(185,262)(186,266)(187,270)(188,269)(189,268)(190,267)(191,271)
(192,275)(193,274)(194,273)(195,272)(196,276)(197,280)(198,279)(199,278)
(200,277)(201,281)(202,285)(203,284)(204,283)(205,282)(206,286)(207,290)
(208,289)(209,288)(210,287)(211,291)(212,295)(213,294)(214,293)(215,292)
(216,296)(217,300)(218,299)(219,298)(220,297)(221,301)(222,305)(223,304)
(224,303)(225,302)(226,306)(227,310)(228,309)(229,308)(230,307)(231,311)
(232,315)(233,314)(234,313)(235,312)(236,316)(237,320)(238,319)(239,318)
(240,317);
s3 := Sym(320)!(  1,202)(  2,201)(  3,205)(  4,204)(  5,203)(  6,207)(  7,206)
(  8,210)(  9,209)( 10,208)( 11,212)( 12,211)( 13,215)( 14,214)( 15,213)
( 16,217)( 17,216)( 18,220)( 19,219)( 20,218)( 21,222)( 22,221)( 23,225)
( 24,224)( 25,223)( 26,227)( 27,226)( 28,230)( 29,229)( 30,228)( 31,232)
( 32,231)( 33,235)( 34,234)( 35,233)( 36,237)( 37,236)( 38,240)( 39,239)
( 40,238)( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)
( 48,170)( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)
( 56,177)( 57,176)( 58,180)( 59,179)( 60,178)( 61,182)( 62,181)( 63,185)
( 64,184)( 65,183)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,192)
( 72,191)( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)
( 80,198)( 81,287)( 82,286)( 83,290)( 84,289)( 85,288)( 86,282)( 87,281)
( 88,285)( 89,284)( 90,283)( 91,297)( 92,296)( 93,300)( 94,299)( 95,298)
( 96,292)( 97,291)( 98,295)( 99,294)(100,293)(101,307)(102,306)(103,310)
(104,309)(105,308)(106,302)(107,301)(108,305)(109,304)(110,303)(111,317)
(112,316)(113,320)(114,319)(115,318)(116,312)(117,311)(118,315)(119,314)
(120,313)(121,247)(122,246)(123,250)(124,249)(125,248)(126,242)(127,241)
(128,245)(129,244)(130,243)(131,257)(132,256)(133,260)(134,259)(135,258)
(136,252)(137,251)(138,255)(139,254)(140,253)(141,267)(142,266)(143,270)
(144,269)(145,268)(146,262)(147,261)(148,265)(149,264)(150,263)(151,277)
(152,276)(153,280)(154,279)(155,278)(156,272)(157,271)(158,275)(159,274)
(160,273);
poly := sub<Sym(320)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope