Polytope of Type {2,80,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,80,4}*1280a
if this polytope has a name.
Group : SmallGroup(1280,323306)
Rank : 4
Schlafli Type : {2,80,4}
Number of vertices, edges, etc : 2, 80, 160, 4
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 : {2,40,4}*640a, {2,80,2}*640
   4-fold quotients : {2,20,4}*320, {2,40,2}*320
   5-fold quotients : {2,16,4}*256a
   8-fold quotients : {2,20,2}*160, {2,10,4}*160
   10-fold quotients : {2,8,4}*128a, {2,16,2}*128
   16-fold quotients : {2,10,2}*80
   20-fold quotients : {2,4,4}*64, {2,8,2}*64
   32-fold quotients : {2,5,2}*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 := (  3,163)(  4,167)(  5,166)(  6,165)(  7,164)(  8,168)(  9,172)( 10,171)
( 11,170)( 12,169)( 13,178)( 14,182)( 15,181)( 16,180)( 17,179)( 18,173)
( 19,177)( 20,176)( 21,175)( 22,174)( 23,183)( 24,187)( 25,186)( 26,185)
( 27,184)( 28,188)( 29,192)( 30,191)( 31,190)( 32,189)( 33,198)( 34,202)
( 35,201)( 36,200)( 37,199)( 38,193)( 39,197)( 40,196)( 41,195)( 42,194)
( 43,213)( 44,217)( 45,216)( 46,215)( 47,214)( 48,218)( 49,222)( 50,221)
( 51,220)( 52,219)( 53,203)( 54,207)( 55,206)( 56,205)( 57,204)( 58,208)
( 59,212)( 60,211)( 61,210)( 62,209)( 63,233)( 64,237)( 65,236)( 66,235)
( 67,234)( 68,238)( 69,242)( 70,241)( 71,240)( 72,239)( 73,223)( 74,227)
( 75,226)( 76,225)( 77,224)( 78,228)( 79,232)( 80,231)( 81,230)( 82,229)
( 83,243)( 84,247)( 85,246)( 86,245)( 87,244)( 88,248)( 89,252)( 90,251)
( 91,250)( 92,249)( 93,258)( 94,262)( 95,261)( 96,260)( 97,259)( 98,253)
( 99,257)(100,256)(101,255)(102,254)(103,263)(104,267)(105,266)(106,265)
(107,264)(108,268)(109,272)(110,271)(111,270)(112,269)(113,278)(114,282)
(115,281)(116,280)(117,279)(118,273)(119,277)(120,276)(121,275)(122,274)
(123,293)(124,297)(125,296)(126,295)(127,294)(128,298)(129,302)(130,301)
(131,300)(132,299)(133,283)(134,287)(135,286)(136,285)(137,284)(138,288)
(139,292)(140,291)(141,290)(142,289)(143,313)(144,317)(145,316)(146,315)
(147,314)(148,318)(149,322)(150,321)(151,320)(152,319)(153,303)(154,307)
(155,306)(156,305)(157,304)(158,308)(159,312)(160,311)(161,310)(162,309);;
s2 := (  3,  4)(  5,  7)(  8,  9)( 10, 12)( 13, 19)( 14, 18)( 15, 22)( 16, 21)
( 17, 20)( 23, 24)( 25, 27)( 28, 29)( 30, 32)( 33, 39)( 34, 38)( 35, 42)
( 36, 41)( 37, 40)( 43, 54)( 44, 53)( 45, 57)( 46, 56)( 47, 55)( 48, 59)
( 49, 58)( 50, 62)( 51, 61)( 52, 60)( 63, 74)( 64, 73)( 65, 77)( 66, 76)
( 67, 75)( 68, 79)( 69, 78)( 70, 82)( 71, 81)( 72, 80)( 83,104)( 84,103)
( 85,107)( 86,106)( 87,105)( 88,109)( 89,108)( 90,112)( 91,111)( 92,110)
( 93,119)( 94,118)( 95,122)( 96,121)( 97,120)( 98,114)( 99,113)(100,117)
(101,116)(102,115)(123,154)(124,153)(125,157)(126,156)(127,155)(128,159)
(129,158)(130,162)(131,161)(132,160)(133,144)(134,143)(135,147)(136,146)
(137,145)(138,149)(139,148)(140,152)(141,151)(142,150)(163,204)(164,203)
(165,207)(166,206)(167,205)(168,209)(169,208)(170,212)(171,211)(172,210)
(173,219)(174,218)(175,222)(176,221)(177,220)(178,214)(179,213)(180,217)
(181,216)(182,215)(183,224)(184,223)(185,227)(186,226)(187,225)(188,229)
(189,228)(190,232)(191,231)(192,230)(193,239)(194,238)(195,242)(196,241)
(197,240)(198,234)(199,233)(200,237)(201,236)(202,235)(243,304)(244,303)
(245,307)(246,306)(247,305)(248,309)(249,308)(250,312)(251,311)(252,310)
(253,319)(254,318)(255,322)(256,321)(257,320)(258,314)(259,313)(260,317)
(261,316)(262,315)(263,284)(264,283)(265,287)(266,286)(267,285)(268,289)
(269,288)(270,292)(271,291)(272,290)(273,299)(274,298)(275,302)(276,301)
(277,300)(278,294)(279,293)(280,297)(281,296)(282,295);;
s3 := (  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)( 81,161)( 82,162)
(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)(241,321)(242,322);;
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(322)!(1,2);
s1 := Sym(322)!(  3,163)(  4,167)(  5,166)(  6,165)(  7,164)(  8,168)(  9,172)
( 10,171)( 11,170)( 12,169)( 13,178)( 14,182)( 15,181)( 16,180)( 17,179)
( 18,173)( 19,177)( 20,176)( 21,175)( 22,174)( 23,183)( 24,187)( 25,186)
( 26,185)( 27,184)( 28,188)( 29,192)( 30,191)( 31,190)( 32,189)( 33,198)
( 34,202)( 35,201)( 36,200)( 37,199)( 38,193)( 39,197)( 40,196)( 41,195)
( 42,194)( 43,213)( 44,217)( 45,216)( 46,215)( 47,214)( 48,218)( 49,222)
( 50,221)( 51,220)( 52,219)( 53,203)( 54,207)( 55,206)( 56,205)( 57,204)
( 58,208)( 59,212)( 60,211)( 61,210)( 62,209)( 63,233)( 64,237)( 65,236)
( 66,235)( 67,234)( 68,238)( 69,242)( 70,241)( 71,240)( 72,239)( 73,223)
( 74,227)( 75,226)( 76,225)( 77,224)( 78,228)( 79,232)( 80,231)( 81,230)
( 82,229)( 83,243)( 84,247)( 85,246)( 86,245)( 87,244)( 88,248)( 89,252)
( 90,251)( 91,250)( 92,249)( 93,258)( 94,262)( 95,261)( 96,260)( 97,259)
( 98,253)( 99,257)(100,256)(101,255)(102,254)(103,263)(104,267)(105,266)
(106,265)(107,264)(108,268)(109,272)(110,271)(111,270)(112,269)(113,278)
(114,282)(115,281)(116,280)(117,279)(118,273)(119,277)(120,276)(121,275)
(122,274)(123,293)(124,297)(125,296)(126,295)(127,294)(128,298)(129,302)
(130,301)(131,300)(132,299)(133,283)(134,287)(135,286)(136,285)(137,284)
(138,288)(139,292)(140,291)(141,290)(142,289)(143,313)(144,317)(145,316)
(146,315)(147,314)(148,318)(149,322)(150,321)(151,320)(152,319)(153,303)
(154,307)(155,306)(156,305)(157,304)(158,308)(159,312)(160,311)(161,310)
(162,309);
s2 := Sym(322)!(  3,  4)(  5,  7)(  8,  9)( 10, 12)( 13, 19)( 14, 18)( 15, 22)
( 16, 21)( 17, 20)( 23, 24)( 25, 27)( 28, 29)( 30, 32)( 33, 39)( 34, 38)
( 35, 42)( 36, 41)( 37, 40)( 43, 54)( 44, 53)( 45, 57)( 46, 56)( 47, 55)
( 48, 59)( 49, 58)( 50, 62)( 51, 61)( 52, 60)( 63, 74)( 64, 73)( 65, 77)
( 66, 76)( 67, 75)( 68, 79)( 69, 78)( 70, 82)( 71, 81)( 72, 80)( 83,104)
( 84,103)( 85,107)( 86,106)( 87,105)( 88,109)( 89,108)( 90,112)( 91,111)
( 92,110)( 93,119)( 94,118)( 95,122)( 96,121)( 97,120)( 98,114)( 99,113)
(100,117)(101,116)(102,115)(123,154)(124,153)(125,157)(126,156)(127,155)
(128,159)(129,158)(130,162)(131,161)(132,160)(133,144)(134,143)(135,147)
(136,146)(137,145)(138,149)(139,148)(140,152)(141,151)(142,150)(163,204)
(164,203)(165,207)(166,206)(167,205)(168,209)(169,208)(170,212)(171,211)
(172,210)(173,219)(174,218)(175,222)(176,221)(177,220)(178,214)(179,213)
(180,217)(181,216)(182,215)(183,224)(184,223)(185,227)(186,226)(187,225)
(188,229)(189,228)(190,232)(191,231)(192,230)(193,239)(194,238)(195,242)
(196,241)(197,240)(198,234)(199,233)(200,237)(201,236)(202,235)(243,304)
(244,303)(245,307)(246,306)(247,305)(248,309)(249,308)(250,312)(251,311)
(252,310)(253,319)(254,318)(255,322)(256,321)(257,320)(258,314)(259,313)
(260,317)(261,316)(262,315)(263,284)(264,283)(265,287)(266,286)(267,285)
(268,289)(269,288)(270,292)(271,291)(272,290)(273,299)(274,298)(275,302)
(276,301)(277,300)(278,294)(279,293)(280,297)(281,296)(282,295);
s3 := 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)( 43,123)( 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)( 81,161)
( 82,162)(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)(241,321)
(242,322);
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope