Polytope of Type {4,20,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20,4}*640
Also Known As : {{4,20|2},{20,4|2}}. if this polytope has another name.
Group : SmallGroup(640,7665)
Rank : 4
Schlafli Type : {4,20,4}
Number of vertices, edges, etc : 4, 40, 40, 4
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,20,4,2} of size 1280
Vertex Figure Of :
   {2,4,20,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,20,4}*320, {4,20,2}*320, {4,10,4}*320
   4-fold quotients : {2,20,2}*160, {2,10,4}*160, {4,10,2}*160
   5-fold quotients : {4,4,4}*128
   8-fold quotients : {2,10,2}*80
   10-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   16-fold quotients : {2,5,2}*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 : {4,20,8}*1280a, {8,20,4}*1280a, {4,20,8}*1280b, {8,20,4}*1280b, {4,40,4}*1280a, {4,20,4}*1280a, {4,20,4}*1280b, {4,40,4}*1280b, {4,40,4}*1280c, {4,40,4}*1280d
   3-fold covers : {4,60,4}*1920a, {4,20,12}*1920, {12,20,4}*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 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 41, 61)( 42, 65)( 43, 64)( 44, 63)( 45, 62)( 46, 66)( 47, 70)( 48, 69)
( 49, 68)( 50, 67)( 51, 71)( 52, 75)( 53, 74)( 54, 73)( 55, 72)( 56, 76)
( 57, 80)( 58, 79)( 59, 78)( 60, 77)( 81, 91)( 82, 95)( 83, 94)( 84, 93)
( 85, 92)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)(101,111)(102,115)
(103,114)(104,113)(105,112)(106,116)(107,120)(108,119)(109,118)(110,117)
(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)(161,181)(162,185)(163,184)(164,183)
(165,182)(166,186)(167,190)(168,189)(169,188)(170,187)(171,191)(172,195)
(173,194)(174,193)(175,192)(176,196)(177,200)(178,199)(179,198)(180,197)
(202,205)(203,204)(207,210)(208,209)(212,215)(213,214)(217,220)(218,219)
(222,225)(223,224)(227,230)(228,229)(232,235)(233,234)(237,240)(238,239)
(241,271)(242,275)(243,274)(244,273)(245,272)(246,276)(247,280)(248,279)
(249,278)(250,277)(251,261)(252,265)(253,264)(254,263)(255,262)(256,266)
(257,270)(258,269)(259,268)(260,267)(281,291)(282,295)(283,294)(284,293)
(285,292)(286,296)(287,300)(288,299)(289,298)(290,297)(301,311)(302,315)
(303,314)(304,313)(305,312)(306,316)(307,320)(308,319)(309,318)(310,317);;
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);;
s3 := (  1,201)(  2,202)(  3,203)(  4,204)(  5,205)(  6,206)(  7,207)(  8,208)
(  9,209)( 10,210)( 11,211)( 12,212)( 13,213)( 14,214)( 15,215)( 16,216)
( 17,217)( 18,218)( 19,219)( 20,220)( 21,221)( 22,222)( 23,223)( 24,224)
( 25,225)( 26,226)( 27,227)( 28,228)( 29,229)( 30,230)( 31,231)( 32,232)
( 33,233)( 34,234)( 35,235)( 36,236)( 37,237)( 38,238)( 39,239)( 40,240)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)
( 81,286)( 82,287)( 83,288)( 84,289)( 85,290)( 86,281)( 87,282)( 88,283)
( 89,284)( 90,285)( 91,296)( 92,297)( 93,298)( 94,299)( 95,300)( 96,291)
( 97,292)( 98,293)( 99,294)(100,295)(101,306)(102,307)(103,308)(104,309)
(105,310)(106,301)(107,302)(108,303)(109,304)(110,305)(111,316)(112,317)
(113,318)(114,319)(115,320)(116,311)(117,312)(118,313)(119,314)(120,315)
(121,246)(122,247)(123,248)(124,249)(125,250)(126,241)(127,242)(128,243)
(129,244)(130,245)(131,256)(132,257)(133,258)(134,259)(135,260)(136,251)
(137,252)(138,253)(139,254)(140,255)(141,266)(142,267)(143,268)(144,269)
(145,270)(146,261)(147,262)(148,263)(149,264)(150,265)(151,276)(152,277)
(153,278)(154,279)(155,280)(156,271)(157,272)(158,273)(159,274)(160,275);;
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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, 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)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 41, 61)( 42, 65)( 43, 64)( 44, 63)( 45, 62)( 46, 66)( 47, 70)
( 48, 69)( 49, 68)( 50, 67)( 51, 71)( 52, 75)( 53, 74)( 54, 73)( 55, 72)
( 56, 76)( 57, 80)( 58, 79)( 59, 78)( 60, 77)( 81, 91)( 82, 95)( 83, 94)
( 84, 93)( 85, 92)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)(101,111)
(102,115)(103,114)(104,113)(105,112)(106,116)(107,120)(108,119)(109,118)
(110,117)(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)(161,181)(162,185)(163,184)
(164,183)(165,182)(166,186)(167,190)(168,189)(169,188)(170,187)(171,191)
(172,195)(173,194)(174,193)(175,192)(176,196)(177,200)(178,199)(179,198)
(180,197)(202,205)(203,204)(207,210)(208,209)(212,215)(213,214)(217,220)
(218,219)(222,225)(223,224)(227,230)(228,229)(232,235)(233,234)(237,240)
(238,239)(241,271)(242,275)(243,274)(244,273)(245,272)(246,276)(247,280)
(248,279)(249,278)(250,277)(251,261)(252,265)(253,264)(254,263)(255,262)
(256,266)(257,270)(258,269)(259,268)(260,267)(281,291)(282,295)(283,294)
(284,293)(285,292)(286,296)(287,300)(288,299)(289,298)(290,297)(301,311)
(302,315)(303,314)(304,313)(305,312)(306,316)(307,320)(308,319)(309,318)
(310,317);
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);
s3 := Sym(320)!(  1,201)(  2,202)(  3,203)(  4,204)(  5,205)(  6,206)(  7,207)
(  8,208)(  9,209)( 10,210)( 11,211)( 12,212)( 13,213)( 14,214)( 15,215)
( 16,216)( 17,217)( 18,218)( 19,219)( 20,220)( 21,221)( 22,222)( 23,223)
( 24,224)( 25,225)( 26,226)( 27,227)( 28,228)( 29,229)( 30,230)( 31,231)
( 32,232)( 33,233)( 34,234)( 35,235)( 36,236)( 37,237)( 38,238)( 39,239)
( 40,240)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)
( 80,200)( 81,286)( 82,287)( 83,288)( 84,289)( 85,290)( 86,281)( 87,282)
( 88,283)( 89,284)( 90,285)( 91,296)( 92,297)( 93,298)( 94,299)( 95,300)
( 96,291)( 97,292)( 98,293)( 99,294)(100,295)(101,306)(102,307)(103,308)
(104,309)(105,310)(106,301)(107,302)(108,303)(109,304)(110,305)(111,316)
(112,317)(113,318)(114,319)(115,320)(116,311)(117,312)(118,313)(119,314)
(120,315)(121,246)(122,247)(123,248)(124,249)(125,250)(126,241)(127,242)
(128,243)(129,244)(130,245)(131,256)(132,257)(133,258)(134,259)(135,260)
(136,251)(137,252)(138,253)(139,254)(140,255)(141,266)(142,267)(143,268)
(144,269)(145,270)(146,261)(147,262)(148,263)(149,264)(150,265)(151,276)
(152,277)(153,278)(154,279)(155,280)(156,271)(157,272)(158,273)(159,274)
(160,275);
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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