Polytope of Type {4,8,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,10}*640b
if this polytope has a name.
Group : SmallGroup(640,14089)
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
4-fold quotients : {2,4,10}*160, {4,2,10}*160
5-fold quotients : {4,8,2}*128b
8-fold quotients : {4,2,5}*80, {2,2,10}*80
10-fold quotients : {4,4,2}*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}*1280c, {8,8,10}*1280d, {4,8,20}*1280b
3-fold covers : {4,8,30}*1920b, {12,8,10}*1920b, {4,24,10}*1920b
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)( 21,106)( 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);;
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)(241,276)(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);;
s2 := ( 1,201)( 2,205)( 3,204)( 4,203)( 5,202)( 6,206)( 7,210)( 8,209)
( 9,208)( 10,207)( 11,216)( 12,220)( 13,219)( 14,218)( 15,217)( 16,211)
( 17,215)( 18,214)( 19,213)( 20,212)( 21,226)( 22,230)( 23,229)( 24,228)
( 25,227)( 26,221)( 27,225)( 28,224)( 29,223)( 30,222)( 31,231)( 32,235)
( 33,234)( 34,233)( 35,232)( 36,236)( 37,240)( 38,239)( 39,238)( 40,237)
( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)( 48,169)
( 49,168)( 50,167)( 51,176)( 52,180)( 53,179)( 54,178)( 55,177)( 56,171)
( 57,175)( 58,174)( 59,173)( 60,172)( 61,186)( 62,190)( 63,189)( 64,188)
( 65,187)( 66,181)( 67,185)( 68,184)( 69,183)( 70,182)( 71,191)( 72,195)
( 73,194)( 74,193)( 75,192)( 76,196)( 77,200)( 78,199)( 79,198)( 80,197)
( 81,281)( 82,285)( 83,284)( 84,283)( 85,282)( 86,286)( 87,290)( 88,289)
( 89,288)( 90,287)( 91,296)( 92,300)( 93,299)( 94,298)( 95,297)( 96,291)
( 97,295)( 98,294)( 99,293)(100,292)(101,306)(102,310)(103,309)(104,308)
(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,311)(112,315)
(113,314)(114,313)(115,312)(116,316)(117,320)(118,319)(119,318)(120,317)
(121,241)(122,245)(123,244)(124,243)(125,242)(126,246)(127,250)(128,249)
(129,248)(130,247)(131,256)(132,260)(133,259)(134,258)(135,257)(136,251)
(137,255)(138,254)(139,253)(140,252)(141,266)(142,270)(143,269)(144,268)
(145,267)(146,261)(147,265)(148,264)(149,263)(150,262)(151,271)(152,275)
(153,274)(154,273)(155,272)(156,276)(157,280)(158,279)(159,278)(160,277);;
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*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*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, 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);
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)(241,276)(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);
s2 := Sym(320)!( 1,201)( 2,205)( 3,204)( 4,203)( 5,202)( 6,206)( 7,210)
( 8,209)( 9,208)( 10,207)( 11,216)( 12,220)( 13,219)( 14,218)( 15,217)
( 16,211)( 17,215)( 18,214)( 19,213)( 20,212)( 21,226)( 22,230)( 23,229)
( 24,228)( 25,227)( 26,221)( 27,225)( 28,224)( 29,223)( 30,222)( 31,231)
( 32,235)( 33,234)( 34,233)( 35,232)( 36,236)( 37,240)( 38,239)( 39,238)
( 40,237)( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)
( 48,169)( 49,168)( 50,167)( 51,176)( 52,180)( 53,179)( 54,178)( 55,177)
( 56,171)( 57,175)( 58,174)( 59,173)( 60,172)( 61,186)( 62,190)( 63,189)
( 64,188)( 65,187)( 66,181)( 67,185)( 68,184)( 69,183)( 70,182)( 71,191)
( 72,195)( 73,194)( 74,193)( 75,192)( 76,196)( 77,200)( 78,199)( 79,198)
( 80,197)( 81,281)( 82,285)( 83,284)( 84,283)( 85,282)( 86,286)( 87,290)
( 88,289)( 89,288)( 90,287)( 91,296)( 92,300)( 93,299)( 94,298)( 95,297)
( 96,291)( 97,295)( 98,294)( 99,293)(100,292)(101,306)(102,310)(103,309)
(104,308)(105,307)(106,301)(107,305)(108,304)(109,303)(110,302)(111,311)
(112,315)(113,314)(114,313)(115,312)(116,316)(117,320)(118,319)(119,318)
(120,317)(121,241)(122,245)(123,244)(124,243)(125,242)(126,246)(127,250)
(128,249)(129,248)(130,247)(131,256)(132,260)(133,259)(134,258)(135,257)
(136,251)(137,255)(138,254)(139,253)(140,252)(141,266)(142,270)(143,269)
(144,268)(145,267)(146,261)(147,265)(148,264)(149,263)(150,262)(151,271)
(152,275)(153,274)(154,273)(155,272)(156,276)(157,280)(158,279)(159,278)
(160,277);
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*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*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