Polytope of Type {80,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {80,4}*640a
Also Known As : {80,4|2}. if this polytope has another name.
Group : SmallGroup(640,1983)
Rank : 3
Schlafli Type : {80,4}
Number of vertices, edges, etc : 80, 160, 4
Order of s0s1s2 : 80
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{80,4,2} of size 1280
Vertex Figure Of :
{2,80,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {40,4}*320a, {80,2}*320
4-fold quotients : {20,4}*160, {40,2}*160
5-fold quotients : {16,4}*128a
8-fold quotients : {20,2}*80, {10,4}*80
10-fold quotients : {8,4}*64a, {16,2}*64
16-fold quotients : {10,2}*40
20-fold quotients : {4,4}*32, {8,2}*32
32-fold quotients : {5,2}*20
40-fold quotients : {2,4}*16, {4,2}*16
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {80,4}*1280a, {80,8}*1280c, {80,8}*1280d, {160,4}*1280a, {160,4}*1280b
3-fold covers : {240,4}*1920a, {80,12}*1920a
Permutation Representation (GAP) :
s0 := ( 1,161)( 2,165)( 3,164)( 4,163)( 5,162)( 6,166)( 7,170)( 8,169)
( 9,168)( 10,167)( 11,176)( 12,180)( 13,179)( 14,178)( 15,177)( 16,171)
( 17,175)( 18,174)( 19,173)( 20,172)( 21,181)( 22,185)( 23,184)( 24,183)
( 25,182)( 26,186)( 27,190)( 28,189)( 29,188)( 30,187)( 31,196)( 32,200)
( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)( 40,192)
( 41,211)( 42,215)( 43,214)( 44,213)( 45,212)( 46,216)( 47,220)( 48,219)
( 49,218)( 50,217)( 51,201)( 52,205)( 53,204)( 54,203)( 55,202)( 56,206)
( 57,210)( 58,209)( 59,208)( 60,207)( 61,231)( 62,235)( 63,234)( 64,233)
( 65,232)( 66,236)( 67,240)( 68,239)( 69,238)( 70,237)( 71,221)( 72,225)
( 73,224)( 74,223)( 75,222)( 76,226)( 77,230)( 78,229)( 79,228)( 80,227)
( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,256)( 92,260)( 93,259)( 94,258)( 95,257)( 96,251)
( 97,255)( 98,254)( 99,253)(100,252)(101,261)(102,265)(103,264)(104,263)
(105,262)(106,266)(107,270)(108,269)(109,268)(110,267)(111,276)(112,280)
(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)(120,272)
(121,291)(122,295)(123,294)(124,293)(125,292)(126,296)(127,300)(128,299)
(129,298)(130,297)(131,281)(132,285)(133,284)(134,283)(135,282)(136,286)
(137,290)(138,289)(139,288)(140,287)(141,311)(142,315)(143,314)(144,313)
(145,312)(146,316)(147,320)(148,319)(149,318)(150,317)(151,301)(152,305)
(153,304)(154,303)(155,302)(156,306)(157,310)(158,309)(159,308)(160,307);;
s1 := ( 1, 2)( 3, 5)( 6, 7)( 8, 10)( 11, 17)( 12, 16)( 13, 20)( 14, 19)
( 15, 18)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 37)( 32, 36)( 33, 40)
( 34, 39)( 35, 38)( 41, 52)( 42, 51)( 43, 55)( 44, 54)( 45, 53)( 46, 57)
( 47, 56)( 48, 60)( 49, 59)( 50, 58)( 61, 72)( 62, 71)( 63, 75)( 64, 74)
( 65, 73)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 81,102)( 82,101)
( 83,105)( 84,104)( 85,103)( 86,107)( 87,106)( 88,110)( 89,109)( 90,108)
( 91,117)( 92,116)( 93,120)( 94,119)( 95,118)( 96,112)( 97,111)( 98,115)
( 99,114)(100,113)(121,152)(122,151)(123,155)(124,154)(125,153)(126,157)
(127,156)(128,160)(129,159)(130,158)(131,142)(132,141)(133,145)(134,144)
(135,143)(136,147)(137,146)(138,150)(139,149)(140,148)(161,202)(162,201)
(163,205)(164,204)(165,203)(166,207)(167,206)(168,210)(169,209)(170,208)
(171,217)(172,216)(173,220)(174,219)(175,218)(176,212)(177,211)(178,215)
(179,214)(180,213)(181,222)(182,221)(183,225)(184,224)(185,223)(186,227)
(187,226)(188,230)(189,229)(190,228)(191,237)(192,236)(193,240)(194,239)
(195,238)(196,232)(197,231)(198,235)(199,234)(200,233)(241,302)(242,301)
(243,305)(244,304)(245,303)(246,307)(247,306)(248,310)(249,309)(250,308)
(251,317)(252,316)(253,320)(254,319)(255,318)(256,312)(257,311)(258,315)
(259,314)(260,313)(261,282)(262,281)(263,285)(264,284)(265,283)(266,287)
(267,286)(268,290)(269,289)(270,288)(271,297)(272,296)(273,300)(274,299)
(275,298)(276,292)(277,291)(278,295)(279,294)(280,293);;
s2 := ( 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]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!( 1,161)( 2,165)( 3,164)( 4,163)( 5,162)( 6,166)( 7,170)
( 8,169)( 9,168)( 10,167)( 11,176)( 12,180)( 13,179)( 14,178)( 15,177)
( 16,171)( 17,175)( 18,174)( 19,173)( 20,172)( 21,181)( 22,185)( 23,184)
( 24,183)( 25,182)( 26,186)( 27,190)( 28,189)( 29,188)( 30,187)( 31,196)
( 32,200)( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)
( 40,192)( 41,211)( 42,215)( 43,214)( 44,213)( 45,212)( 46,216)( 47,220)
( 48,219)( 49,218)( 50,217)( 51,201)( 52,205)( 53,204)( 54,203)( 55,202)
( 56,206)( 57,210)( 58,209)( 59,208)( 60,207)( 61,231)( 62,235)( 63,234)
( 64,233)( 65,232)( 66,236)( 67,240)( 68,239)( 69,238)( 70,237)( 71,221)
( 72,225)( 73,224)( 74,223)( 75,222)( 76,226)( 77,230)( 78,229)( 79,228)
( 80,227)( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)
( 88,249)( 89,248)( 90,247)( 91,256)( 92,260)( 93,259)( 94,258)( 95,257)
( 96,251)( 97,255)( 98,254)( 99,253)(100,252)(101,261)(102,265)(103,264)
(104,263)(105,262)(106,266)(107,270)(108,269)(109,268)(110,267)(111,276)
(112,280)(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)
(120,272)(121,291)(122,295)(123,294)(124,293)(125,292)(126,296)(127,300)
(128,299)(129,298)(130,297)(131,281)(132,285)(133,284)(134,283)(135,282)
(136,286)(137,290)(138,289)(139,288)(140,287)(141,311)(142,315)(143,314)
(144,313)(145,312)(146,316)(147,320)(148,319)(149,318)(150,317)(151,301)
(152,305)(153,304)(154,303)(155,302)(156,306)(157,310)(158,309)(159,308)
(160,307);
s1 := Sym(320)!( 1, 2)( 3, 5)( 6, 7)( 8, 10)( 11, 17)( 12, 16)( 13, 20)
( 14, 19)( 15, 18)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 37)( 32, 36)
( 33, 40)( 34, 39)( 35, 38)( 41, 52)( 42, 51)( 43, 55)( 44, 54)( 45, 53)
( 46, 57)( 47, 56)( 48, 60)( 49, 59)( 50, 58)( 61, 72)( 62, 71)( 63, 75)
( 64, 74)( 65, 73)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 81,102)
( 82,101)( 83,105)( 84,104)( 85,103)( 86,107)( 87,106)( 88,110)( 89,109)
( 90,108)( 91,117)( 92,116)( 93,120)( 94,119)( 95,118)( 96,112)( 97,111)
( 98,115)( 99,114)(100,113)(121,152)(122,151)(123,155)(124,154)(125,153)
(126,157)(127,156)(128,160)(129,159)(130,158)(131,142)(132,141)(133,145)
(134,144)(135,143)(136,147)(137,146)(138,150)(139,149)(140,148)(161,202)
(162,201)(163,205)(164,204)(165,203)(166,207)(167,206)(168,210)(169,209)
(170,208)(171,217)(172,216)(173,220)(174,219)(175,218)(176,212)(177,211)
(178,215)(179,214)(180,213)(181,222)(182,221)(183,225)(184,224)(185,223)
(186,227)(187,226)(188,230)(189,229)(190,228)(191,237)(192,236)(193,240)
(194,239)(195,238)(196,232)(197,231)(198,235)(199,234)(200,233)(241,302)
(242,301)(243,305)(244,304)(245,303)(246,307)(247,306)(248,310)(249,309)
(250,308)(251,317)(252,316)(253,320)(254,319)(255,318)(256,312)(257,311)
(258,315)(259,314)(260,313)(261,282)(262,281)(263,285)(264,284)(265,283)
(266,287)(267,286)(268,290)(269,289)(270,288)(271,297)(272,296)(273,300)
(274,299)(275,298)(276,292)(277,291)(278,295)(279,294)(280,293);
s2 := 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>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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