Polytope of Type {4,8,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,10}*640a
Also Known As : {{4,8|2},{8,10|2}}. if this polytope has another name.
Group : SmallGroup(640,13813)
Rank : 4
Schlafli Type : {4,8,10}
Number of vertices, edges, etc : 4, 16, 40, 10
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,8,10,2} of size 1280
Vertex Figure Of :
   {2,4,8,10} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4,10}*320, {2,8,10}*320
   4-fold quotients : {2,4,10}*160, {4,2,10}*160
   5-fold quotients : {4,8,2}*128a
   8-fold quotients : {4,2,5}*80, {2,2,10}*80
   10-fold quotients : {4,4,2}*64, {2,8,2}*64
   16-fold quotients : {2,2,5}*40
   20-fold quotients : {2,4,2}*32, {4,2,2}*32
   40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8,10}*1280a, {8,8,10}*1280a, {8,8,10}*1280b, {4,8,20}*1280d, {4,16,10}*1280a, {4,16,10}*1280b
   3-fold covers : {4,8,30}*1920a, {12,8,10}*1920a, {4,24,10}*1920a
Permutation Representation (GAP) : 
s0 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)( 24,104)
( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)
( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)
(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)
(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)
(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)
(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)
(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288)
(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)(216,296)
(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)
(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)
(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320);;
s1 := ( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 71, 76)
( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81, 91)( 82, 92)( 83, 93)( 84, 94)
( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,116)(102,117)
(103,118)(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)(110,115)
(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)
(129,139)(130,140)(141,156)(142,157)(143,158)(144,159)(145,160)(146,151)
(147,152)(148,153)(149,154)(150,155)(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)
(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)
(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)
(217,237)(218,238)(219,239)(220,240)(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)
(281,311)(282,312)(283,313)(284,314)(285,315)(286,316)(287,317)(288,318)
(289,319)(290,320)(291,301)(292,302)(293,303)(294,304)(295,305)(296,306)
(297,307)(298,308)(299,309)(300,310);;
s2 := (  1,201)(  2,205)(  3,204)(  4,203)(  5,202)(  6,206)(  7,210)(  8,209)
(  9,208)( 10,207)( 11,211)( 12,215)( 13,214)( 14,213)( 15,212)( 16,216)
( 17,220)( 18,219)( 19,218)( 20,217)( 21,226)( 22,230)( 23,229)( 24,228)
( 25,227)( 26,221)( 27,225)( 28,224)( 29,223)( 30,222)( 31,236)( 32,240)
( 33,239)( 34,238)( 35,237)( 36,231)( 37,235)( 38,234)( 39,233)( 40,232)
( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)( 48,169)
( 49,168)( 50,167)( 51,171)( 52,175)( 53,174)( 54,173)( 55,172)( 56,176)
( 57,180)( 58,179)( 59,178)( 60,177)( 61,186)( 62,190)( 63,189)( 64,188)
( 65,187)( 66,181)( 67,185)( 68,184)( 69,183)( 70,182)( 71,196)( 72,200)
( 73,199)( 74,198)( 75,197)( 76,191)( 77,195)( 78,194)( 79,193)( 80,192)
( 81,281)( 82,285)( 83,284)( 84,283)( 85,282)( 86,286)( 87,290)( 88,289)
( 89,288)( 90,287)( 91,291)( 92,295)( 93,294)( 94,293)( 95,292)( 96,296)
( 97,300)( 98,299)( 99,298)(100,297)(101,306)(102,310)(103,309)(104,308)
(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,316)(112,320)
(113,319)(114,318)(115,317)(116,311)(117,315)(118,314)(119,313)(120,312)
(121,241)(122,245)(123,244)(124,243)(125,242)(126,246)(127,250)(128,249)
(129,248)(130,247)(131,251)(132,255)(133,254)(134,253)(135,252)(136,256)
(137,260)(138,259)(139,258)(140,257)(141,266)(142,270)(143,269)(144,268)
(145,267)(146,261)(147,265)(148,264)(149,263)(150,262)(151,276)(152,280)
(153,279)(154,278)(155,277)(156,271)(157,275)(158,274)(159,273)(160,272);;
s3 := (  1, 42)(  2, 41)(  3, 45)(  4, 44)(  5, 43)(  6, 47)(  7, 46)(  8, 50)
(  9, 49)( 10, 48)( 11, 52)( 12, 51)( 13, 55)( 14, 54)( 15, 53)( 16, 57)
( 17, 56)( 18, 60)( 19, 59)( 20, 58)( 21, 62)( 22, 61)( 23, 65)( 24, 64)
( 25, 63)( 26, 67)( 27, 66)( 28, 70)( 29, 69)( 30, 68)( 31, 72)( 32, 71)
( 33, 75)( 34, 74)( 35, 73)( 36, 77)( 37, 76)( 38, 80)( 39, 79)( 40, 78)
( 81,122)( 82,121)( 83,125)( 84,124)( 85,123)( 86,127)( 87,126)( 88,130)
( 89,129)( 90,128)( 91,132)( 92,131)( 93,135)( 94,134)( 95,133)( 96,137)
( 97,136)( 98,140)( 99,139)(100,138)(101,142)(102,141)(103,145)(104,144)
(105,143)(106,147)(107,146)(108,150)(109,149)(110,148)(111,152)(112,151)
(113,155)(114,154)(115,153)(116,157)(117,156)(118,160)(119,159)(120,158)
(161,202)(162,201)(163,205)(164,204)(165,203)(166,207)(167,206)(168,210)
(169,209)(170,208)(171,212)(172,211)(173,215)(174,214)(175,213)(176,217)
(177,216)(178,220)(179,219)(180,218)(181,222)(182,221)(183,225)(184,224)
(185,223)(186,227)(187,226)(188,230)(189,229)(190,228)(191,232)(192,231)
(193,235)(194,234)(195,233)(196,237)(197,236)(198,240)(199,239)(200,238)
(241,282)(242,281)(243,285)(244,284)(245,283)(246,287)(247,286)(248,290)
(249,289)(250,288)(251,292)(252,291)(253,295)(254,294)(255,293)(256,297)
(257,296)(258,300)(259,299)(260,298)(261,302)(262,301)(263,305)(264,304)
(265,303)(266,307)(267,306)(268,310)(269,309)(270,308)(271,312)(272,311)
(273,315)(274,314)(275,313)(276,317)(277,316)(278,320)(279,319)(280,318);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)
( 24,104)( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)
( 48,128)( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)
( 56,136)( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)
( 72,152)( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)
( 80,160)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)
(168,248)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)
(176,256)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)
(184,264)(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)
(192,272)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)
(200,280)(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)
(208,288)(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)
(216,296)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)
(224,304)(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)
(232,312)(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)
(240,320);
s1 := Sym(320)!( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)
( 71, 76)( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81, 91)( 82, 92)( 83, 93)
( 84, 94)( 85, 95)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(101,116)
(102,117)(103,118)(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)
(110,115)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)
(128,138)(129,139)(130,140)(141,156)(142,157)(143,158)(144,159)(145,160)
(146,151)(147,152)(148,153)(149,154)(150,155)(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)(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)
(208,228)(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)
(216,236)(217,237)(218,238)(219,239)(220,240)(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)(281,311)(282,312)(283,313)(284,314)(285,315)(286,316)(287,317)
(288,318)(289,319)(290,320)(291,301)(292,302)(293,303)(294,304)(295,305)
(296,306)(297,307)(298,308)(299,309)(300,310);
s2 := Sym(320)!(  1,201)(  2,205)(  3,204)(  4,203)(  5,202)(  6,206)(  7,210)
(  8,209)(  9,208)( 10,207)( 11,211)( 12,215)( 13,214)( 14,213)( 15,212)
( 16,216)( 17,220)( 18,219)( 19,218)( 20,217)( 21,226)( 22,230)( 23,229)
( 24,228)( 25,227)( 26,221)( 27,225)( 28,224)( 29,223)( 30,222)( 31,236)
( 32,240)( 33,239)( 34,238)( 35,237)( 36,231)( 37,235)( 38,234)( 39,233)
( 40,232)( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)
( 48,169)( 49,168)( 50,167)( 51,171)( 52,175)( 53,174)( 54,173)( 55,172)
( 56,176)( 57,180)( 58,179)( 59,178)( 60,177)( 61,186)( 62,190)( 63,189)
( 64,188)( 65,187)( 66,181)( 67,185)( 68,184)( 69,183)( 70,182)( 71,196)
( 72,200)( 73,199)( 74,198)( 75,197)( 76,191)( 77,195)( 78,194)( 79,193)
( 80,192)( 81,281)( 82,285)( 83,284)( 84,283)( 85,282)( 86,286)( 87,290)
( 88,289)( 89,288)( 90,287)( 91,291)( 92,295)( 93,294)( 94,293)( 95,292)
( 96,296)( 97,300)( 98,299)( 99,298)(100,297)(101,306)(102,310)(103,309)
(104,308)(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,316)
(112,320)(113,319)(114,318)(115,317)(116,311)(117,315)(118,314)(119,313)
(120,312)(121,241)(122,245)(123,244)(124,243)(125,242)(126,246)(127,250)
(128,249)(129,248)(130,247)(131,251)(132,255)(133,254)(134,253)(135,252)
(136,256)(137,260)(138,259)(139,258)(140,257)(141,266)(142,270)(143,269)
(144,268)(145,267)(146,261)(147,265)(148,264)(149,263)(150,262)(151,276)
(152,280)(153,279)(154,278)(155,277)(156,271)(157,275)(158,274)(159,273)
(160,272);
s3 := Sym(320)!(  1, 42)(  2, 41)(  3, 45)(  4, 44)(  5, 43)(  6, 47)(  7, 46)
(  8, 50)(  9, 49)( 10, 48)( 11, 52)( 12, 51)( 13, 55)( 14, 54)( 15, 53)
( 16, 57)( 17, 56)( 18, 60)( 19, 59)( 20, 58)( 21, 62)( 22, 61)( 23, 65)
( 24, 64)( 25, 63)( 26, 67)( 27, 66)( 28, 70)( 29, 69)( 30, 68)( 31, 72)
( 32, 71)( 33, 75)( 34, 74)( 35, 73)( 36, 77)( 37, 76)( 38, 80)( 39, 79)
( 40, 78)( 81,122)( 82,121)( 83,125)( 84,124)( 85,123)( 86,127)( 87,126)
( 88,130)( 89,129)( 90,128)( 91,132)( 92,131)( 93,135)( 94,134)( 95,133)
( 96,137)( 97,136)( 98,140)( 99,139)(100,138)(101,142)(102,141)(103,145)
(104,144)(105,143)(106,147)(107,146)(108,150)(109,149)(110,148)(111,152)
(112,151)(113,155)(114,154)(115,153)(116,157)(117,156)(118,160)(119,159)
(120,158)(161,202)(162,201)(163,205)(164,204)(165,203)(166,207)(167,206)
(168,210)(169,209)(170,208)(171,212)(172,211)(173,215)(174,214)(175,213)
(176,217)(177,216)(178,220)(179,219)(180,218)(181,222)(182,221)(183,225)
(184,224)(185,223)(186,227)(187,226)(188,230)(189,229)(190,228)(191,232)
(192,231)(193,235)(194,234)(195,233)(196,237)(197,236)(198,240)(199,239)
(200,238)(241,282)(242,281)(243,285)(244,284)(245,283)(246,287)(247,286)
(248,290)(249,289)(250,288)(251,292)(252,291)(253,295)(254,294)(255,293)
(256,297)(257,296)(258,300)(259,299)(260,298)(261,302)(262,301)(263,305)
(264,304)(265,303)(266,307)(267,306)(268,310)(269,309)(270,308)(271,312)
(272,311)(273,315)(274,314)(275,313)(276,317)(277,316)(278,320)(279,319)
(280,318);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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