Polytope of Type {8,4,10}

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