Polytope of Type {2,4,80}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,80}*1280b
if this polytope has a name.
Group : SmallGroup(1280,323454)
Rank : 4
Schlafli Type : {2,4,80}
Number of vertices, edges, etc : 2, 4, 160, 80
Order of s₀s₁s₂s₃ : 80
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,40}*640a
   4-fold quotients : {2,4,20}*320, {2,2,40}*320
   5-fold quotients : {2,4,16}*256b
   8-fold quotients : {2,2,20}*160, {2,4,10}*160
   10-fold quotients : {2,4,8}*128a
   16-fold quotients : {2,2,10}*80
   20-fold quotients : {2,4,4}*64, {2,2,8}*64
   32-fold quotients : {2,2,5}*40
   40-fold quotients : {2,2,4}*32, {2,4,2}*32
   80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  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)
( 43,128)( 44,129)( 45,130)( 46,131)( 47,132)( 48,123)( 49,124)( 50,125)
( 51,126)( 52,127)( 53,138)( 54,139)( 55,140)( 56,141)( 57,142)( 58,133)
( 59,134)( 60,135)( 61,136)( 62,137)( 63,148)( 64,149)( 65,150)( 66,151)
( 67,152)( 68,143)( 69,144)( 70,145)( 71,146)( 72,147)( 73,158)( 74,159)
( 75,160)( 76,161)( 77,162)( 78,153)( 79,154)( 80,155)( 81,156)( 82,157)
(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)
(203,288)(204,289)(205,290)(206,291)(207,292)(208,283)(209,284)(210,285)
(211,286)(212,287)(213,298)(214,299)(215,300)(216,301)(217,302)(218,293)
(219,294)(220,295)(221,296)(222,297)(223,308)(224,309)(225,310)(226,311)
(227,312)(228,303)(229,304)(230,305)(231,306)(232,307)(233,318)(234,319)
(235,320)(236,321)(237,322)(238,313)(239,314)(240,315)(241,316)(242,317);;
s2 := (  4,  7)(  5,  6)(  9, 12)( 10, 11)( 13, 18)( 14, 22)( 15, 21)( 16, 20)
( 17, 19)( 24, 27)( 25, 26)( 29, 32)( 30, 31)( 33, 38)( 34, 42)( 35, 41)
( 36, 40)( 37, 39)( 43, 53)( 44, 57)( 45, 56)( 46, 55)( 47, 54)( 48, 58)
( 49, 62)( 50, 61)( 51, 60)( 52, 59)( 63, 73)( 64, 77)( 65, 76)( 66, 75)
( 67, 74)( 68, 78)( 69, 82)( 70, 81)( 71, 80)( 72, 79)( 83,103)( 84,107)
( 85,106)( 86,105)( 87,104)( 88,108)( 89,112)( 90,111)( 91,110)( 92,109)
( 93,118)( 94,122)( 95,121)( 96,120)( 97,119)( 98,113)( 99,117)(100,116)
(101,115)(102,114)(123,153)(124,157)(125,156)(126,155)(127,154)(128,158)
(129,162)(130,161)(131,160)(132,159)(133,143)(134,147)(135,146)(136,145)
(137,144)(138,148)(139,152)(140,151)(141,150)(142,149)(163,203)(164,207)
(165,206)(166,205)(167,204)(168,208)(169,212)(170,211)(171,210)(172,209)
(173,218)(174,222)(175,221)(176,220)(177,219)(178,213)(179,217)(180,216)
(181,215)(182,214)(183,223)(184,227)(185,226)(186,225)(187,224)(188,228)
(189,232)(190,231)(191,230)(192,229)(193,238)(194,242)(195,241)(196,240)
(197,239)(198,233)(199,237)(200,236)(201,235)(202,234)(243,308)(244,312)
(245,311)(246,310)(247,309)(248,303)(249,307)(250,306)(251,305)(252,304)
(253,313)(254,317)(255,316)(256,315)(257,314)(258,318)(259,322)(260,321)
(261,320)(262,319)(263,288)(264,292)(265,291)(266,290)(267,289)(268,283)
(269,287)(270,286)(271,285)(272,284)(273,293)(274,297)(275,296)(276,295)
(277,294)(278,298)(279,302)(280,301)(281,300)(282,299);;
s3 := (  3,164)(  4,163)(  5,167)(  6,166)(  7,165)(  8,169)(  9,168)( 10,172)
( 11,171)( 12,170)( 13,179)( 14,178)( 15,182)( 16,181)( 17,180)( 18,174)
( 19,173)( 20,177)( 21,176)( 22,175)( 23,189)( 24,188)( 25,192)( 26,191)
( 27,190)( 28,184)( 29,183)( 30,187)( 31,186)( 32,185)( 33,194)( 34,193)
( 35,197)( 36,196)( 37,195)( 38,199)( 39,198)( 40,202)( 41,201)( 42,200)
( 43,214)( 44,213)( 45,217)( 46,216)( 47,215)( 48,219)( 49,218)( 50,222)
( 51,221)( 52,220)( 53,204)( 54,203)( 55,207)( 56,206)( 57,205)( 58,209)
( 59,208)( 60,212)( 61,211)( 62,210)( 63,239)( 64,238)( 65,242)( 66,241)
( 67,240)( 68,234)( 69,233)( 70,237)( 71,236)( 72,235)( 73,229)( 74,228)
( 75,232)( 76,231)( 77,230)( 78,224)( 79,223)( 80,227)( 81,226)( 82,225)
( 83,244)( 84,243)( 85,247)( 86,246)( 87,245)( 88,249)( 89,248)( 90,252)
( 91,251)( 92,250)( 93,259)( 94,258)( 95,262)( 96,261)( 97,260)( 98,254)
( 99,253)(100,257)(101,256)(102,255)(103,269)(104,268)(105,272)(106,271)
(107,270)(108,264)(109,263)(110,267)(111,266)(112,265)(113,274)(114,273)
(115,277)(116,276)(117,275)(118,279)(119,278)(120,282)(121,281)(122,280)
(123,294)(124,293)(125,297)(126,296)(127,295)(128,299)(129,298)(130,302)
(131,301)(132,300)(133,284)(134,283)(135,287)(136,286)(137,285)(138,289)
(139,288)(140,292)(141,291)(142,290)(143,319)(144,318)(145,322)(146,321)
(147,320)(148,314)(149,313)(150,317)(151,316)(152,315)(153,309)(154,308)
(155,312)(156,311)(157,310)(158,304)(159,303)(160,307)(161,306)(162,305);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2, 
s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(322)!(1,2);
s1 := Sym(322)!(  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)( 43,128)( 44,129)( 45,130)( 46,131)( 47,132)( 48,123)( 49,124)
( 50,125)( 51,126)( 52,127)( 53,138)( 54,139)( 55,140)( 56,141)( 57,142)
( 58,133)( 59,134)( 60,135)( 61,136)( 62,137)( 63,148)( 64,149)( 65,150)
( 66,151)( 67,152)( 68,143)( 69,144)( 70,145)( 71,146)( 72,147)( 73,158)
( 74,159)( 75,160)( 76,161)( 77,162)( 78,153)( 79,154)( 80,155)( 81,156)
( 82,157)(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)(203,288)(204,289)(205,290)(206,291)(207,292)(208,283)(209,284)
(210,285)(211,286)(212,287)(213,298)(214,299)(215,300)(216,301)(217,302)
(218,293)(219,294)(220,295)(221,296)(222,297)(223,308)(224,309)(225,310)
(226,311)(227,312)(228,303)(229,304)(230,305)(231,306)(232,307)(233,318)
(234,319)(235,320)(236,321)(237,322)(238,313)(239,314)(240,315)(241,316)
(242,317);
s2 := Sym(322)!(  4,  7)(  5,  6)(  9, 12)( 10, 11)( 13, 18)( 14, 22)( 15, 21)
( 16, 20)( 17, 19)( 24, 27)( 25, 26)( 29, 32)( 30, 31)( 33, 38)( 34, 42)
( 35, 41)( 36, 40)( 37, 39)( 43, 53)( 44, 57)( 45, 56)( 46, 55)( 47, 54)
( 48, 58)( 49, 62)( 50, 61)( 51, 60)( 52, 59)( 63, 73)( 64, 77)( 65, 76)
( 66, 75)( 67, 74)( 68, 78)( 69, 82)( 70, 81)( 71, 80)( 72, 79)( 83,103)
( 84,107)( 85,106)( 86,105)( 87,104)( 88,108)( 89,112)( 90,111)( 91,110)
( 92,109)( 93,118)( 94,122)( 95,121)( 96,120)( 97,119)( 98,113)( 99,117)
(100,116)(101,115)(102,114)(123,153)(124,157)(125,156)(126,155)(127,154)
(128,158)(129,162)(130,161)(131,160)(132,159)(133,143)(134,147)(135,146)
(136,145)(137,144)(138,148)(139,152)(140,151)(141,150)(142,149)(163,203)
(164,207)(165,206)(166,205)(167,204)(168,208)(169,212)(170,211)(171,210)
(172,209)(173,218)(174,222)(175,221)(176,220)(177,219)(178,213)(179,217)
(180,216)(181,215)(182,214)(183,223)(184,227)(185,226)(186,225)(187,224)
(188,228)(189,232)(190,231)(191,230)(192,229)(193,238)(194,242)(195,241)
(196,240)(197,239)(198,233)(199,237)(200,236)(201,235)(202,234)(243,308)
(244,312)(245,311)(246,310)(247,309)(248,303)(249,307)(250,306)(251,305)
(252,304)(253,313)(254,317)(255,316)(256,315)(257,314)(258,318)(259,322)
(260,321)(261,320)(262,319)(263,288)(264,292)(265,291)(266,290)(267,289)
(268,283)(269,287)(270,286)(271,285)(272,284)(273,293)(274,297)(275,296)
(276,295)(277,294)(278,298)(279,302)(280,301)(281,300)(282,299);
s3 := Sym(322)!(  3,164)(  4,163)(  5,167)(  6,166)(  7,165)(  8,169)(  9,168)
( 10,172)( 11,171)( 12,170)( 13,179)( 14,178)( 15,182)( 16,181)( 17,180)
( 18,174)( 19,173)( 20,177)( 21,176)( 22,175)( 23,189)( 24,188)( 25,192)
( 26,191)( 27,190)( 28,184)( 29,183)( 30,187)( 31,186)( 32,185)( 33,194)
( 34,193)( 35,197)( 36,196)( 37,195)( 38,199)( 39,198)( 40,202)( 41,201)
( 42,200)( 43,214)( 44,213)( 45,217)( 46,216)( 47,215)( 48,219)( 49,218)
( 50,222)( 51,221)( 52,220)( 53,204)( 54,203)( 55,207)( 56,206)( 57,205)
( 58,209)( 59,208)( 60,212)( 61,211)( 62,210)( 63,239)( 64,238)( 65,242)
( 66,241)( 67,240)( 68,234)( 69,233)( 70,237)( 71,236)( 72,235)( 73,229)
( 74,228)( 75,232)( 76,231)( 77,230)( 78,224)( 79,223)( 80,227)( 81,226)
( 82,225)( 83,244)( 84,243)( 85,247)( 86,246)( 87,245)( 88,249)( 89,248)
( 90,252)( 91,251)( 92,250)( 93,259)( 94,258)( 95,262)( 96,261)( 97,260)
( 98,254)( 99,253)(100,257)(101,256)(102,255)(103,269)(104,268)(105,272)
(106,271)(107,270)(108,264)(109,263)(110,267)(111,266)(112,265)(113,274)
(114,273)(115,277)(116,276)(117,275)(118,279)(119,278)(120,282)(121,281)
(122,280)(123,294)(124,293)(125,297)(126,296)(127,295)(128,299)(129,298)
(130,302)(131,301)(132,300)(133,284)(134,283)(135,287)(136,286)(137,285)
(138,289)(139,288)(140,292)(141,291)(142,290)(143,319)(144,318)(145,322)
(146,321)(147,320)(148,314)(149,313)(150,317)(151,316)(152,315)(153,309)
(154,308)(155,312)(156,311)(157,310)(158,304)(159,303)(160,307)(161,306)
(162,305);
poly := sub<Sym(322)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2, 
s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2 >;

to this polytope