Polytope of Type {2,8,40}

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