Polytope of Type {8,10,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,10,4}*640
Also Known As : {{8,10|2},{10,4|2}}. if this polytope has another name.
Group : SmallGroup(640,13837)
Rank : 4
Schlafli Type : {8,10,4}
Number of vertices, edges, etc : 8, 40, 20, 4
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,10,4,2} of size 1280
Vertex Figure Of :
   {2,8,10,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,10,4}*320, {8,10,2}*320
   4-fold quotients : {2,10,4}*160, {4,10,2}*160
   5-fold quotients : {8,2,4}*128
   8-fold quotients : {2,10,2}*80
   10-fold quotients : {4,2,4}*64, {8,2,2}*64
   16-fold quotients : {2,5,2}*40
   20-fold quotients : {2,2,4}*32, {4,2,2}*32
   40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,10,8}*1280, {8,20,4}*1280a, {16,10,4}*1280
   3-fold covers : {8,30,4}*1920a, {8,10,12}*1920, {24,10,4}*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, 66)( 22, 67)( 23, 68)( 24, 69)
( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)( 32, 77)
( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)
( 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)(101,146)(102,147)(103,148)(104,149)
(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)
(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155)
(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)(181,226)(182,227)(183,228)(184,229)
(185,230)(186,221)(187,222)(188,223)(189,224)(190,225)(191,236)(192,237)
(193,238)(194,239)(195,240)(196,231)(197,232)(198,233)(199,234)(200,235)
(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)(261,306)(262,307)(263,308)(264,309)
(265,310)(266,301)(267,302)(268,303)(269,304)(270,305)(271,316)(272,317)
(273,318)(274,319)(275,320)(276,311)(277,312)(278,313)(279,314)(280,315);;
s1 := (  1,161)(  2,165)(  3,164)(  4,163)(  5,162)(  6,166)(  7,170)(  8,169)
(  9,168)( 10,167)( 11,171)( 12,175)( 13,174)( 14,173)( 15,172)( 16,176)
( 17,180)( 18,179)( 19,178)( 20,177)( 21,186)( 22,190)( 23,189)( 24,188)
( 25,187)( 26,181)( 27,185)( 28,184)( 29,183)( 30,182)( 31,196)( 32,200)
( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)( 40,192)
( 41,221)( 42,225)( 43,224)( 44,223)( 45,222)( 46,226)( 47,230)( 48,229)
( 49,228)( 50,227)( 51,231)( 52,235)( 53,234)( 54,233)( 55,232)( 56,236)
( 57,240)( 58,239)( 59,238)( 60,237)( 61,201)( 62,205)( 63,204)( 64,203)
( 65,202)( 66,206)( 67,210)( 68,209)( 69,208)( 70,207)( 71,211)( 72,215)
( 73,214)( 74,213)( 75,212)( 76,216)( 77,220)( 78,219)( 79,218)( 80,217)
( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,251)( 92,255)( 93,254)( 94,253)( 95,252)( 96,256)
( 97,260)( 98,259)( 99,258)(100,257)(101,266)(102,270)(103,269)(104,268)
(105,267)(106,261)(107,265)(108,264)(109,263)(110,262)(111,276)(112,280)
(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)(120,272)
(121,301)(122,305)(123,304)(124,303)(125,302)(126,306)(127,310)(128,309)
(129,308)(130,307)(131,311)(132,315)(133,314)(134,313)(135,312)(136,316)
(137,320)(138,319)(139,318)(140,317)(141,281)(142,285)(143,284)(144,283)
(145,282)(146,286)(147,290)(148,289)(149,288)(150,287)(151,291)(152,295)
(153,294)(154,293)(155,292)(156,296)(157,300)(158,299)(159,298)(160,297);;
s2 := (  1, 62)(  2, 61)(  3, 65)(  4, 64)(  5, 63)(  6, 67)(  7, 66)(  8, 70)
(  9, 69)( 10, 68)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)( 16, 77)
( 17, 76)( 18, 80)( 19, 79)( 20, 78)( 21, 47)( 22, 46)( 23, 50)( 24, 49)
( 25, 48)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 31, 57)( 32, 56)
( 33, 60)( 34, 59)( 35, 58)( 36, 52)( 37, 51)( 38, 55)( 39, 54)( 40, 53)
( 81,152)( 82,151)( 83,155)( 84,154)( 85,153)( 86,157)( 87,156)( 88,160)
( 89,159)( 90,158)( 91,142)( 92,141)( 93,145)( 94,144)( 95,143)( 96,147)
( 97,146)( 98,150)( 99,149)(100,148)(101,137)(102,136)(103,140)(104,139)
(105,138)(106,132)(107,131)(108,135)(109,134)(110,133)(111,127)(112,126)
(113,130)(114,129)(115,128)(116,122)(117,121)(118,125)(119,124)(120,123)
(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)
(241,292)(242,291)(243,295)(244,294)(245,293)(246,297)(247,296)(248,300)
(249,299)(250,298)(251,282)(252,281)(253,285)(254,284)(255,283)(256,287)
(257,286)(258,290)(259,289)(260,288)(261,312)(262,311)(263,315)(264,314)
(265,313)(266,317)(267,316)(268,320)(269,319)(270,318)(271,302)(272,301)
(273,305)(274,304)(275,303)(276,307)(277,306)(278,310)(279,309)(280,308);;
s3 := (  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);;
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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 ];;
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, 66)( 22, 67)( 23, 68)
( 24, 69)( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)
( 32, 77)( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)
( 40, 75)( 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)(101,146)(102,147)(103,148)
(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)
(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)
(120,155)(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)(181,226)(182,227)(183,228)
(184,229)(185,230)(186,221)(187,222)(188,223)(189,224)(190,225)(191,236)
(192,237)(193,238)(194,239)(195,240)(196,231)(197,232)(198,233)(199,234)
(200,235)(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)(261,306)(262,307)(263,308)
(264,309)(265,310)(266,301)(267,302)(268,303)(269,304)(270,305)(271,316)
(272,317)(273,318)(274,319)(275,320)(276,311)(277,312)(278,313)(279,314)
(280,315);
s1 := Sym(320)!(  1,161)(  2,165)(  3,164)(  4,163)(  5,162)(  6,166)(  7,170)
(  8,169)(  9,168)( 10,167)( 11,171)( 12,175)( 13,174)( 14,173)( 15,172)
( 16,176)( 17,180)( 18,179)( 19,178)( 20,177)( 21,186)( 22,190)( 23,189)
( 24,188)( 25,187)( 26,181)( 27,185)( 28,184)( 29,183)( 30,182)( 31,196)
( 32,200)( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)
( 40,192)( 41,221)( 42,225)( 43,224)( 44,223)( 45,222)( 46,226)( 47,230)
( 48,229)( 49,228)( 50,227)( 51,231)( 52,235)( 53,234)( 54,233)( 55,232)
( 56,236)( 57,240)( 58,239)( 59,238)( 60,237)( 61,201)( 62,205)( 63,204)
( 64,203)( 65,202)( 66,206)( 67,210)( 68,209)( 69,208)( 70,207)( 71,211)
( 72,215)( 73,214)( 74,213)( 75,212)( 76,216)( 77,220)( 78,219)( 79,218)
( 80,217)( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)
( 88,249)( 89,248)( 90,247)( 91,251)( 92,255)( 93,254)( 94,253)( 95,252)
( 96,256)( 97,260)( 98,259)( 99,258)(100,257)(101,266)(102,270)(103,269)
(104,268)(105,267)(106,261)(107,265)(108,264)(109,263)(110,262)(111,276)
(112,280)(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)
(120,272)(121,301)(122,305)(123,304)(124,303)(125,302)(126,306)(127,310)
(128,309)(129,308)(130,307)(131,311)(132,315)(133,314)(134,313)(135,312)
(136,316)(137,320)(138,319)(139,318)(140,317)(141,281)(142,285)(143,284)
(144,283)(145,282)(146,286)(147,290)(148,289)(149,288)(150,287)(151,291)
(152,295)(153,294)(154,293)(155,292)(156,296)(157,300)(158,299)(159,298)
(160,297);
s2 := Sym(320)!(  1, 62)(  2, 61)(  3, 65)(  4, 64)(  5, 63)(  6, 67)(  7, 66)
(  8, 70)(  9, 69)( 10, 68)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)
( 16, 77)( 17, 76)( 18, 80)( 19, 79)( 20, 78)( 21, 47)( 22, 46)( 23, 50)
( 24, 49)( 25, 48)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 31, 57)
( 32, 56)( 33, 60)( 34, 59)( 35, 58)( 36, 52)( 37, 51)( 38, 55)( 39, 54)
( 40, 53)( 81,152)( 82,151)( 83,155)( 84,154)( 85,153)( 86,157)( 87,156)
( 88,160)( 89,159)( 90,158)( 91,142)( 92,141)( 93,145)( 94,144)( 95,143)
( 96,147)( 97,146)( 98,150)( 99,149)(100,148)(101,137)(102,136)(103,140)
(104,139)(105,138)(106,132)(107,131)(108,135)(109,134)(110,133)(111,127)
(112,126)(113,130)(114,129)(115,128)(116,122)(117,121)(118,125)(119,124)
(120,123)(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)(241,292)(242,291)(243,295)(244,294)(245,293)(246,297)(247,296)
(248,300)(249,299)(250,298)(251,282)(252,281)(253,285)(254,284)(255,283)
(256,287)(257,286)(258,290)(259,289)(260,288)(261,312)(262,311)(263,315)
(264,314)(265,313)(266,317)(267,316)(268,320)(269,319)(270,318)(271,302)
(272,301)(273,305)(274,304)(275,303)(276,307)(277,306)(278,310)(279,309)
(280,308);
s3 := 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);
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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 >;
References : None.
to this polytope