Polytope of Type {10,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,8,4}*640b
if this polytope has a name.
Group : SmallGroup(640,14089)
Rank : 4
Schlafli Type : {10,8,4}
Number of vertices, edges, etc : 10, 40, 16, 4
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,8,4,2} of size 1280
Vertex Figure Of :
{2,10,8,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,4,4}*320
4-fold quotients : {10,2,4}*160, {10,4,2}*160
5-fold quotients : {2,8,4}*128b
8-fold quotients : {5,2,4}*80, {10,2,2}*80
10-fold quotients : {2,4,4}*64
16-fold quotients : {5,2,2}*40
20-fold quotients : {2,2,4}*32, {2,4,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,8,4}*1280a, {10,8,8}*1280a, {10,8,8}*1280d, {20,8,4}*1280b
3-fold covers : {30,8,4}*1920b, {10,8,12}*1920b, {10,24,4}*1920b
Permutation Representation (GAP) :
s0 := ( 1, 41)( 2, 45)( 3, 44)( 4, 43)( 5, 42)( 6, 46)( 7, 50)( 8, 49)
( 9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 56)
( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)( 24, 63)
( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)( 32, 75)
( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)
( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)
( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)
( 97,140)( 98,139)( 99,138)(100,137)(101,141)(102,145)(103,144)(104,143)
(105,142)(106,146)(107,150)(108,149)(109,148)(110,147)(111,151)(112,155)
(113,154)(114,153)(115,152)(116,156)(117,160)(118,159)(119,158)(120,157)
(161,201)(162,205)(163,204)(164,203)(165,202)(166,206)(167,210)(168,209)
(169,208)(170,207)(171,211)(172,215)(173,214)(174,213)(175,212)(176,216)
(177,220)(178,219)(179,218)(180,217)(181,221)(182,225)(183,224)(184,223)
(185,222)(186,226)(187,230)(188,229)(189,228)(190,227)(191,231)(192,235)
(193,234)(194,233)(195,232)(196,236)(197,240)(198,239)(199,238)(200,237)
(241,281)(242,285)(243,284)(244,283)(245,282)(246,286)(247,290)(248,289)
(249,288)(250,287)(251,291)(252,295)(253,294)(254,293)(255,292)(256,296)
(257,300)(258,299)(259,298)(260,297)(261,301)(262,305)(263,304)(264,303)
(265,302)(266,306)(267,310)(268,309)(269,308)(270,307)(271,311)(272,315)
(273,314)(274,313)(275,312)(276,316)(277,320)(278,319)(279,318)(280,317);;
s1 := ( 1,202)( 2,201)( 3,205)( 4,204)( 5,203)( 6,207)( 7,206)( 8,210)
( 9,209)( 10,208)( 11,217)( 12,216)( 13,220)( 14,219)( 15,218)( 16,212)
( 17,211)( 18,215)( 19,214)( 20,213)( 21,227)( 22,226)( 23,230)( 24,229)
( 25,228)( 26,222)( 27,221)( 28,225)( 29,224)( 30,223)( 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)( 51,177)( 52,176)( 53,180)( 54,179)( 55,178)( 56,172)
( 57,171)( 58,175)( 59,174)( 60,173)( 61,187)( 62,186)( 63,190)( 64,189)
( 65,188)( 66,182)( 67,181)( 68,185)( 69,184)( 70,183)( 71,192)( 72,191)
( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)( 80,198)
( 81,282)( 82,281)( 83,285)( 84,284)( 85,283)( 86,287)( 87,286)( 88,290)
( 89,289)( 90,288)( 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)(111,312)(112,311)
(113,315)(114,314)(115,313)(116,317)(117,316)(118,320)(119,319)(120,318)
(121,242)(122,241)(123,245)(124,244)(125,243)(126,247)(127,246)(128,250)
(129,249)(130,248)(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)(151,272)(152,271)
(153,275)(154,274)(155,273)(156,277)(157,276)(158,280)(159,279)(160,278);;
s2 := ( 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)(241,276)(242,277)(243,278)(244,279)
(245,280)(246,271)(247,272)(248,273)(249,274)(250,275)(251,266)(252,267)
(253,268)(254,269)(255,270)(256,261)(257,262)(258,263)(259,264)(260,265)
(281,316)(282,317)(283,318)(284,319)(285,320)(286,311)(287,312)(288,313)
(289,314)(290,315)(291,306)(292,307)(293,308)(294,309)(295,310)(296,301)
(297,302)(298,303)(299,304)(300,305);;
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)( 21,106)( 22,107)( 23,108)( 24,109)
( 25,110)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,116)( 32,117)
( 33,118)( 34,119)( 35,120)( 36,111)( 37,112)( 38,113)( 39,114)( 40,115)
( 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)( 61,146)( 62,147)( 63,148)( 64,149)
( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)( 72,157)
( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)
(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)(181,266)(182,267)(183,268)(184,269)
(185,270)(186,261)(187,262)(188,263)(189,264)(190,265)(191,276)(192,277)
(193,278)(194,279)(195,280)(196,271)(197,272)(198,273)(199,274)(200,275)
(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)(221,306)(222,307)(223,308)(224,309)
(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)(232,317)
(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)(240,315);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(320)!( 1, 41)( 2, 45)( 3, 44)( 4, 43)( 5, 42)( 6, 46)( 7, 50)
( 8, 49)( 9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)
( 16, 56)( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)
( 24, 63)( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)
( 32, 75)( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)
( 40, 77)( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)
( 88,129)( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)
( 96,136)( 97,140)( 98,139)( 99,138)(100,137)(101,141)(102,145)(103,144)
(104,143)(105,142)(106,146)(107,150)(108,149)(109,148)(110,147)(111,151)
(112,155)(113,154)(114,153)(115,152)(116,156)(117,160)(118,159)(119,158)
(120,157)(161,201)(162,205)(163,204)(164,203)(165,202)(166,206)(167,210)
(168,209)(169,208)(170,207)(171,211)(172,215)(173,214)(174,213)(175,212)
(176,216)(177,220)(178,219)(179,218)(180,217)(181,221)(182,225)(183,224)
(184,223)(185,222)(186,226)(187,230)(188,229)(189,228)(190,227)(191,231)
(192,235)(193,234)(194,233)(195,232)(196,236)(197,240)(198,239)(199,238)
(200,237)(241,281)(242,285)(243,284)(244,283)(245,282)(246,286)(247,290)
(248,289)(249,288)(250,287)(251,291)(252,295)(253,294)(254,293)(255,292)
(256,296)(257,300)(258,299)(259,298)(260,297)(261,301)(262,305)(263,304)
(264,303)(265,302)(266,306)(267,310)(268,309)(269,308)(270,307)(271,311)
(272,315)(273,314)(274,313)(275,312)(276,316)(277,320)(278,319)(279,318)
(280,317);
s1 := Sym(320)!( 1,202)( 2,201)( 3,205)( 4,204)( 5,203)( 6,207)( 7,206)
( 8,210)( 9,209)( 10,208)( 11,217)( 12,216)( 13,220)( 14,219)( 15,218)
( 16,212)( 17,211)( 18,215)( 19,214)( 20,213)( 21,227)( 22,226)( 23,230)
( 24,229)( 25,228)( 26,222)( 27,221)( 28,225)( 29,224)( 30,223)( 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)( 51,177)( 52,176)( 53,180)( 54,179)( 55,178)
( 56,172)( 57,171)( 58,175)( 59,174)( 60,173)( 61,187)( 62,186)( 63,190)
( 64,189)( 65,188)( 66,182)( 67,181)( 68,185)( 69,184)( 70,183)( 71,192)
( 72,191)( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)
( 80,198)( 81,282)( 82,281)( 83,285)( 84,284)( 85,283)( 86,287)( 87,286)
( 88,290)( 89,289)( 90,288)( 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)(111,312)
(112,311)(113,315)(114,314)(115,313)(116,317)(117,316)(118,320)(119,319)
(120,318)(121,242)(122,241)(123,245)(124,244)(125,243)(126,247)(127,246)
(128,250)(129,249)(130,248)(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)(151,272)
(152,271)(153,275)(154,274)(155,273)(156,277)(157,276)(158,280)(159,279)
(160,278);
s2 := 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)(241,276)(242,277)(243,278)
(244,279)(245,280)(246,271)(247,272)(248,273)(249,274)(250,275)(251,266)
(252,267)(253,268)(254,269)(255,270)(256,261)(257,262)(258,263)(259,264)
(260,265)(281,316)(282,317)(283,318)(284,319)(285,320)(286,311)(287,312)
(288,313)(289,314)(290,315)(291,306)(292,307)(293,308)(294,309)(295,310)
(296,301)(297,302)(298,303)(299,304)(300,305);
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)( 21,106)( 22,107)( 23,108)
( 24,109)( 25,110)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,116)
( 32,117)( 33,118)( 34,119)( 35,120)( 36,111)( 37,112)( 38,113)( 39,114)
( 40,115)( 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)( 61,146)( 62,147)( 63,148)
( 64,149)( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)
( 72,157)( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)
( 80,155)(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)(181,266)(182,267)(183,268)
(184,269)(185,270)(186,261)(187,262)(188,263)(189,264)(190,265)(191,276)
(192,277)(193,278)(194,279)(195,280)(196,271)(197,272)(198,273)(199,274)
(200,275)(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)(221,306)(222,307)(223,308)
(224,309)(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)
(232,317)(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)
(240,315);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope