Polytope of Type {2,40,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,40,4}*1280a
if this polytope has a name.
Group : SmallGroup(1280,141634)
Rank : 4
Schlafli Type : {2,40,4}
Number of vertices, edges, etc : 2, 80, 160, 8
Order of s₀s₁s₂s₃ : 40
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, {2,20,4}*640, {2,40,4}*640b
   4-fold quotients : {2,20,4}*320, {2,40,2}*320
   5-fold quotients : {2,8,4}*256a
   8-fold quotients : {2,20,2}*160, {2,10,4}*160
   10-fold quotients : {2,8,4}*128a, {2,8,4}*128b, {2,4,4}*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)( 13,173)( 14,177)( 15,176)( 16,175)( 17,174)( 18,178)
( 19,182)( 20,181)( 21,180)( 22,179)( 23,188)( 24,192)( 25,191)( 26,190)
( 27,189)( 28,183)( 29,187)( 30,186)( 31,185)( 32,184)( 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)( 63,238)( 64,242)( 65,241)( 66,240)
( 67,239)( 68,233)( 69,237)( 70,236)( 71,235)( 72,234)( 73,228)( 74,232)
( 75,231)( 76,230)( 77,229)( 78,223)( 79,227)( 80,226)( 81,225)( 82,224)
( 83,243)( 84,247)( 85,246)( 86,245)( 87,244)( 88,248)( 89,252)( 90,251)
( 91,250)( 92,249)( 93,253)( 94,257)( 95,256)( 96,255)( 97,254)( 98,258)
( 99,262)(100,261)(101,260)(102,259)(103,268)(104,272)(105,271)(106,270)
(107,269)(108,263)(109,267)(110,266)(111,265)(112,264)(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)(143,318)(144,322)(145,321)(146,320)
(147,319)(148,313)(149,317)(150,316)(151,315)(152,314)(153,308)(154,312)
(155,311)(156,310)(157,309)(158,303)(159,307)(160,306)(161,305)(162,304);;
s2 := (  3,  4)(  5,  7)(  8,  9)( 10, 12)( 13, 14)( 15, 17)( 18, 19)( 20, 22)
( 23, 24)( 25, 27)( 28, 29)( 30, 32)( 33, 34)( 35, 37)( 38, 39)( 40, 42)
( 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)( 93,114)( 94,113)
( 95,117)( 96,116)( 97,115)( 98,119)( 99,118)(100,122)(101,121)(102,120)
(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)(173,214)(174,213)
(175,217)(176,216)(177,215)(178,219)(179,218)(180,222)(181,221)(182,220)
(183,224)(184,223)(185,227)(186,226)(187,225)(188,229)(189,228)(190,232)
(191,231)(192,230)(193,234)(194,233)(195,237)(196,236)(197,235)(198,239)
(199,238)(200,242)(201,241)(202,240)(243,309)(244,308)(245,312)(246,311)
(247,310)(248,304)(249,303)(250,307)(251,306)(252,305)(253,319)(254,318)
(255,322)(256,321)(257,320)(258,314)(259,313)(260,317)(261,316)(262,315)
(263,289)(264,288)(265,292)(266,291)(267,290)(268,284)(269,283)(270,287)
(271,286)(272,285)(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)
( 43,128)( 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);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*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)( 13,173)( 14,177)( 15,176)( 16,175)( 17,174)
( 18,178)( 19,182)( 20,181)( 21,180)( 22,179)( 23,188)( 24,192)( 25,191)
( 26,190)( 27,189)( 28,183)( 29,187)( 30,186)( 31,185)( 32,184)( 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)( 63,238)( 64,242)( 65,241)
( 66,240)( 67,239)( 68,233)( 69,237)( 70,236)( 71,235)( 72,234)( 73,228)
( 74,232)( 75,231)( 76,230)( 77,229)( 78,223)( 79,227)( 80,226)( 81,225)
( 82,224)( 83,243)( 84,247)( 85,246)( 86,245)( 87,244)( 88,248)( 89,252)
( 90,251)( 91,250)( 92,249)( 93,253)( 94,257)( 95,256)( 96,255)( 97,254)
( 98,258)( 99,262)(100,261)(101,260)(102,259)(103,268)(104,272)(105,271)
(106,270)(107,269)(108,263)(109,267)(110,266)(111,265)(112,264)(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)(143,318)(144,322)(145,321)
(146,320)(147,319)(148,313)(149,317)(150,316)(151,315)(152,314)(153,308)
(154,312)(155,311)(156,310)(157,309)(158,303)(159,307)(160,306)(161,305)
(162,304);
s2 := Sym(322)!(  3,  4)(  5,  7)(  8,  9)( 10, 12)( 13, 14)( 15, 17)( 18, 19)
( 20, 22)( 23, 24)( 25, 27)( 28, 29)( 30, 32)( 33, 34)( 35, 37)( 38, 39)
( 40, 42)( 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)( 93,114)
( 94,113)( 95,117)( 96,116)( 97,115)( 98,119)( 99,118)(100,122)(101,121)
(102,120)(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)(173,214)
(174,213)(175,217)(176,216)(177,215)(178,219)(179,218)(180,222)(181,221)
(182,220)(183,224)(184,223)(185,227)(186,226)(187,225)(188,229)(189,228)
(190,232)(191,231)(192,230)(193,234)(194,233)(195,237)(196,236)(197,235)
(198,239)(199,238)(200,242)(201,241)(202,240)(243,309)(244,308)(245,312)
(246,311)(247,310)(248,304)(249,303)(250,307)(251,306)(252,305)(253,319)
(254,318)(255,322)(256,321)(257,320)(258,314)(259,313)(260,317)(261,316)
(262,315)(263,289)(264,288)(265,292)(266,291)(267,290)(268,284)(269,283)
(270,287)(271,286)(272,285)(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)( 43,128)( 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);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*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