Polytope of Type {2,40,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,40,8}*1280a
if this polytope has a name.
Group : SmallGroup(1280,145173)
Rank : 4
Schlafli Type : {2,40,8}
Number of vertices, edges, etc : 2, 40, 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}*640b, {2,20,8}*640a
   4-fold quotients : {2,20,4}*320, {2,10,8}*320
   5-fold quotients : {2,8,8}*256a
   8-fold quotients : {2,20,2}*160, {2,10,4}*160
   10-fold quotients : {2,4,8}*128a, {2,8,4}*128b
   16-fold quotients : {2,10,2}*80
   20-fold quotients : {2,4,4}*64, {2,2,8}*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, 83)(  4, 87)(  5, 86)(  6, 85)(  7, 84)(  8, 88)(  9, 92)( 10, 91)
( 11, 90)( 12, 89)( 13, 93)( 14, 97)( 15, 96)( 16, 95)( 17, 94)( 18, 98)
( 19,102)( 20,101)( 21,100)( 22, 99)( 23,108)( 24,112)( 25,111)( 26,110)
( 27,109)( 28,103)( 29,107)( 30,106)( 31,105)( 32,104)( 33,118)( 34,122)
( 35,121)( 36,120)( 37,119)( 38,113)( 39,117)( 40,116)( 41,115)( 42,114)
( 43,128)( 44,132)( 45,131)( 46,130)( 47,129)( 48,123)( 49,127)( 50,126)
( 51,125)( 52,124)( 53,138)( 54,142)( 55,141)( 56,140)( 57,139)( 58,133)
( 59,137)( 60,136)( 61,135)( 62,134)( 63,143)( 64,147)( 65,146)( 66,145)
( 67,144)( 68,148)( 69,152)( 70,151)( 71,150)( 72,149)( 73,153)( 74,157)
( 75,156)( 76,155)( 77,154)( 78,158)( 79,162)( 80,161)( 81,160)( 82,159)
(163,243)(164,247)(165,246)(166,245)(167,244)(168,248)(169,252)(170,251)
(171,250)(172,249)(173,253)(174,257)(175,256)(176,255)(177,254)(178,258)
(179,262)(180,261)(181,260)(182,259)(183,268)(184,272)(185,271)(186,270)
(187,269)(188,263)(189,267)(190,266)(191,265)(192,264)(193,278)(194,282)
(195,281)(196,280)(197,279)(198,273)(199,277)(200,276)(201,275)(202,274)
(203,288)(204,292)(205,291)(206,290)(207,289)(208,283)(209,287)(210,286)
(211,285)(212,284)(213,298)(214,302)(215,301)(216,300)(217,299)(218,293)
(219,297)(220,296)(221,295)(222,294)(223,303)(224,307)(225,306)(226,305)
(227,304)(228,308)(229,312)(230,311)(231,310)(232,309)(233,313)(234,317)
(235,316)(236,315)(237,314)(238,318)(239,322)(240,321)(241,320)(242,319);;
s2 := (  3,  4)(  5,  7)(  8,  9)( 10, 12)( 13, 14)( 15, 17)( 18, 19)( 20, 22)
( 23, 29)( 24, 28)( 25, 32)( 26, 31)( 27, 30)( 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, 79)( 64, 78)( 65, 82)( 66, 81)
( 67, 80)( 68, 74)( 69, 73)( 70, 77)( 71, 76)( 72, 75)( 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)(183,229)(184,228)(185,232)(186,231)(187,230)(188,224)
(189,223)(190,227)(191,226)(192,225)(193,239)(194,238)(195,242)(196,241)
(197,240)(198,234)(199,233)(200,237)(201,236)(202,235)(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 := (  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)(  9,169)( 10,170)
( 11,171)( 12,172)( 13,173)( 14,174)( 15,175)( 16,176)( 17,177)( 18,178)
( 19,179)( 20,180)( 21,181)( 22,182)( 23,188)( 24,189)( 25,190)( 26,191)
( 27,192)( 28,183)( 29,184)( 30,185)( 31,186)( 32,187)( 33,198)( 34,199)
( 35,200)( 36,201)( 37,202)( 38,193)( 39,194)( 40,195)( 41,196)( 42,197)
( 43,213)( 44,214)( 45,215)( 46,216)( 47,217)( 48,218)( 49,219)( 50,220)
( 51,221)( 52,222)( 53,203)( 54,204)( 55,205)( 56,206)( 57,207)( 58,208)
( 59,209)( 60,210)( 61,211)( 62,212)( 63,238)( 64,239)( 65,240)( 66,241)
( 67,242)( 68,233)( 69,234)( 70,235)( 71,236)( 72,237)( 73,228)( 74,229)
( 75,230)( 76,231)( 77,232)( 78,223)( 79,224)( 80,225)( 81,226)( 82,227)
( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)( 88,248)( 89,249)( 90,250)
( 91,251)( 92,252)( 93,253)( 94,254)( 95,255)( 96,256)( 97,257)( 98,258)
( 99,259)(100,260)(101,261)(102,262)(103,268)(104,269)(105,270)(106,271)
(107,272)(108,263)(109,264)(110,265)(111,266)(112,267)(113,278)(114,279)
(115,280)(116,281)(117,282)(118,273)(119,274)(120,275)(121,276)(122,277)
(123,293)(124,294)(125,295)(126,296)(127,297)(128,298)(129,299)(130,300)
(131,301)(132,302)(133,283)(134,284)(135,285)(136,286)(137,287)(138,288)
(139,289)(140,290)(141,291)(142,292)(143,318)(144,319)(145,320)(146,321)
(147,322)(148,313)(149,314)(150,315)(151,316)(152,317)(153,308)(154,309)
(155,310)(156,311)(157,312)(158,303)(159,304)(160,305)(161,306)(162,307);;
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, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope