Polytope of Type {2,4,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,5}*1280
if this polytope has a name.
Group : SmallGroup(1280,1116454)
Rank : 4
Schlafli Type : {2,4,5}
Number of vertices, edges, etc : 2, 64, 160, 80
Order of s0s1s2s3 : 20
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 : {2,4,5}*640
   4-fold quotients : {2,4,5}*320
   32-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (1,2);;
s1 := (  3,179)(  4,180)(  5,182)(  6,181)(  7,184)(  8,183)(  9,185)( 10,186)
( 11,188)( 12,187)( 13,189)( 14,190)( 15,191)( 16,192)( 17,194)( 18,193)
( 19,164)( 20,163)( 21,165)( 22,166)( 23,167)( 24,168)( 25,170)( 26,169)
( 27,171)( 28,172)( 29,174)( 30,173)( 31,176)( 32,175)( 33,177)( 34,178)
( 35,211)( 36,212)( 37,214)( 38,213)( 39,216)( 40,215)( 41,217)( 42,218)
( 43,220)( 44,219)( 45,221)( 46,222)( 47,223)( 48,224)( 49,226)( 50,225)
( 51,196)( 52,195)( 53,197)( 54,198)( 55,199)( 56,200)( 57,202)( 58,201)
( 59,203)( 60,204)( 61,206)( 62,205)( 63,208)( 64,207)( 65,209)( 66,210)
( 67,243)( 68,244)( 69,246)( 70,245)( 71,248)( 72,247)( 73,249)( 74,250)
( 75,252)( 76,251)( 77,253)( 78,254)( 79,255)( 80,256)( 81,258)( 82,257)
( 83,228)( 84,227)( 85,229)( 86,230)( 87,231)( 88,232)( 89,234)( 90,233)
( 91,235)( 92,236)( 93,238)( 94,237)( 95,240)( 96,239)( 97,241)( 98,242)
( 99,275)(100,276)(101,278)(102,277)(103,280)(104,279)(105,281)(106,282)
(107,284)(108,283)(109,285)(110,286)(111,287)(112,288)(113,290)(114,289)
(115,260)(116,259)(117,261)(118,262)(119,263)(120,264)(121,266)(122,265)
(123,267)(124,268)(125,270)(126,269)(127,272)(128,271)(129,273)(130,274)
(131,307)(132,308)(133,310)(134,309)(135,312)(136,311)(137,313)(138,314)
(139,316)(140,315)(141,317)(142,318)(143,319)(144,320)(145,322)(146,321)
(147,292)(148,291)(149,293)(150,294)(151,295)(152,296)(153,298)(154,297)
(155,299)(156,300)(157,302)(158,301)(159,304)(160,303)(161,305)(162,306);;
s2 := (  3, 35)(  4, 36)(  5, 57)(  6, 58)(  7, 59)(  8, 60)(  9, 49)( 10, 50)
( 11, 44)( 12, 43)( 13, 65)( 14, 66)( 15, 52)( 16, 51)( 17, 41)( 18, 42)
( 19, 48)( 20, 47)( 21, 61)( 22, 62)( 23, 56)( 24, 55)( 25, 37)( 26, 38)
( 27, 39)( 28, 40)( 29, 53)( 30, 54)( 31, 63)( 32, 64)( 33, 45)( 34, 46)
( 67,131)( 68,132)( 69,153)( 70,154)( 71,155)( 72,156)( 73,145)( 74,146)
( 75,140)( 76,139)( 77,161)( 78,162)( 79,148)( 80,147)( 81,137)( 82,138)
( 83,144)( 84,143)( 85,157)( 86,158)( 87,152)( 88,151)( 89,133)( 90,134)
( 91,135)( 92,136)( 93,149)( 94,150)( 95,159)( 96,160)( 97,141)( 98,142)
(101,121)(102,122)(103,123)(104,124)(105,113)(106,114)(107,108)(109,129)
(110,130)(111,116)(112,115)(117,125)(118,126)(119,120)(163,196)(164,195)
(165,218)(166,217)(167,220)(168,219)(169,210)(170,209)(171,203)(172,204)
(173,226)(174,225)(175,211)(176,212)(177,202)(178,201)(179,207)(180,208)
(181,222)(182,221)(183,215)(184,216)(185,198)(186,197)(187,200)(188,199)
(189,214)(190,213)(191,224)(192,223)(193,206)(194,205)(227,292)(228,291)
(229,314)(230,313)(231,316)(232,315)(233,306)(234,305)(235,299)(236,300)
(237,322)(238,321)(239,307)(240,308)(241,298)(242,297)(243,303)(244,304)
(245,318)(246,317)(247,311)(248,312)(249,294)(250,293)(251,296)(252,295)
(253,310)(254,309)(255,320)(256,319)(257,302)(258,301)(259,260)(261,282)
(262,281)(263,284)(264,283)(265,274)(266,273)(269,290)(270,289)(271,275)
(272,276)(277,286)(278,285)(287,288);;
s3 := (  3,131)(  4,132)(  5,150)(  6,149)(  7,151)(  8,152)(  9,138)( 10,137)
( 11,162)( 12,161)( 13,143)( 14,144)( 15,141)( 16,142)( 17,156)( 18,155)
( 19,148)( 20,147)( 21,134)( 22,133)( 23,135)( 24,136)( 25,153)( 26,154)
( 27,146)( 28,145)( 29,160)( 30,159)( 31,158)( 32,157)( 33,140)( 34,139)
( 35, 99)( 36,100)( 37,118)( 38,117)( 39,119)( 40,120)( 41,106)( 42,105)
( 43,130)( 44,129)( 45,111)( 46,112)( 47,109)( 48,110)( 49,124)( 50,123)
( 51,116)( 52,115)( 53,102)( 54,101)( 55,103)( 56,104)( 57,121)( 58,122)
( 59,114)( 60,113)( 61,128)( 62,127)( 63,126)( 64,125)( 65,108)( 66,107)
( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 74)( 75, 98)( 76, 97)( 77, 79)
( 78, 80)( 81, 92)( 82, 91)( 83, 84)( 93, 96)( 94, 95)(163,292)(164,291)
(165,309)(166,310)(167,312)(168,311)(169,297)(170,298)(171,321)(172,322)
(173,304)(174,303)(175,302)(176,301)(177,315)(178,316)(179,307)(180,308)
(181,293)(182,294)(183,296)(184,295)(185,314)(186,313)(187,305)(188,306)
(189,319)(190,320)(191,317)(192,318)(193,299)(194,300)(195,260)(196,259)
(197,277)(198,278)(199,280)(200,279)(201,265)(202,266)(203,289)(204,290)
(205,272)(206,271)(207,270)(208,269)(209,283)(210,284)(211,275)(212,276)
(213,261)(214,262)(215,264)(216,263)(217,282)(218,281)(219,273)(220,274)
(221,287)(222,288)(223,285)(224,286)(225,267)(226,268)(227,228)(229,245)
(230,246)(231,248)(232,247)(235,257)(236,258)(237,240)(238,239)(241,251)
(242,252)(249,250)(253,255)(254,256);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(322)!(1,2);
s1 := Sym(322)!(  3,179)(  4,180)(  5,182)(  6,181)(  7,184)(  8,183)(  9,185)
( 10,186)( 11,188)( 12,187)( 13,189)( 14,190)( 15,191)( 16,192)( 17,194)
( 18,193)( 19,164)( 20,163)( 21,165)( 22,166)( 23,167)( 24,168)( 25,170)
( 26,169)( 27,171)( 28,172)( 29,174)( 30,173)( 31,176)( 32,175)( 33,177)
( 34,178)( 35,211)( 36,212)( 37,214)( 38,213)( 39,216)( 40,215)( 41,217)
( 42,218)( 43,220)( 44,219)( 45,221)( 46,222)( 47,223)( 48,224)( 49,226)
( 50,225)( 51,196)( 52,195)( 53,197)( 54,198)( 55,199)( 56,200)( 57,202)
( 58,201)( 59,203)( 60,204)( 61,206)( 62,205)( 63,208)( 64,207)( 65,209)
( 66,210)( 67,243)( 68,244)( 69,246)( 70,245)( 71,248)( 72,247)( 73,249)
( 74,250)( 75,252)( 76,251)( 77,253)( 78,254)( 79,255)( 80,256)( 81,258)
( 82,257)( 83,228)( 84,227)( 85,229)( 86,230)( 87,231)( 88,232)( 89,234)
( 90,233)( 91,235)( 92,236)( 93,238)( 94,237)( 95,240)( 96,239)( 97,241)
( 98,242)( 99,275)(100,276)(101,278)(102,277)(103,280)(104,279)(105,281)
(106,282)(107,284)(108,283)(109,285)(110,286)(111,287)(112,288)(113,290)
(114,289)(115,260)(116,259)(117,261)(118,262)(119,263)(120,264)(121,266)
(122,265)(123,267)(124,268)(125,270)(126,269)(127,272)(128,271)(129,273)
(130,274)(131,307)(132,308)(133,310)(134,309)(135,312)(136,311)(137,313)
(138,314)(139,316)(140,315)(141,317)(142,318)(143,319)(144,320)(145,322)
(146,321)(147,292)(148,291)(149,293)(150,294)(151,295)(152,296)(153,298)
(154,297)(155,299)(156,300)(157,302)(158,301)(159,304)(160,303)(161,305)
(162,306);
s2 := Sym(322)!(  3, 35)(  4, 36)(  5, 57)(  6, 58)(  7, 59)(  8, 60)(  9, 49)
( 10, 50)( 11, 44)( 12, 43)( 13, 65)( 14, 66)( 15, 52)( 16, 51)( 17, 41)
( 18, 42)( 19, 48)( 20, 47)( 21, 61)( 22, 62)( 23, 56)( 24, 55)( 25, 37)
( 26, 38)( 27, 39)( 28, 40)( 29, 53)( 30, 54)( 31, 63)( 32, 64)( 33, 45)
( 34, 46)( 67,131)( 68,132)( 69,153)( 70,154)( 71,155)( 72,156)( 73,145)
( 74,146)( 75,140)( 76,139)( 77,161)( 78,162)( 79,148)( 80,147)( 81,137)
( 82,138)( 83,144)( 84,143)( 85,157)( 86,158)( 87,152)( 88,151)( 89,133)
( 90,134)( 91,135)( 92,136)( 93,149)( 94,150)( 95,159)( 96,160)( 97,141)
( 98,142)(101,121)(102,122)(103,123)(104,124)(105,113)(106,114)(107,108)
(109,129)(110,130)(111,116)(112,115)(117,125)(118,126)(119,120)(163,196)
(164,195)(165,218)(166,217)(167,220)(168,219)(169,210)(170,209)(171,203)
(172,204)(173,226)(174,225)(175,211)(176,212)(177,202)(178,201)(179,207)
(180,208)(181,222)(182,221)(183,215)(184,216)(185,198)(186,197)(187,200)
(188,199)(189,214)(190,213)(191,224)(192,223)(193,206)(194,205)(227,292)
(228,291)(229,314)(230,313)(231,316)(232,315)(233,306)(234,305)(235,299)
(236,300)(237,322)(238,321)(239,307)(240,308)(241,298)(242,297)(243,303)
(244,304)(245,318)(246,317)(247,311)(248,312)(249,294)(250,293)(251,296)
(252,295)(253,310)(254,309)(255,320)(256,319)(257,302)(258,301)(259,260)
(261,282)(262,281)(263,284)(264,283)(265,274)(266,273)(269,290)(270,289)
(271,275)(272,276)(277,286)(278,285)(287,288);
s3 := Sym(322)!(  3,131)(  4,132)(  5,150)(  6,149)(  7,151)(  8,152)(  9,138)
( 10,137)( 11,162)( 12,161)( 13,143)( 14,144)( 15,141)( 16,142)( 17,156)
( 18,155)( 19,148)( 20,147)( 21,134)( 22,133)( 23,135)( 24,136)( 25,153)
( 26,154)( 27,146)( 28,145)( 29,160)( 30,159)( 31,158)( 32,157)( 33,140)
( 34,139)( 35, 99)( 36,100)( 37,118)( 38,117)( 39,119)( 40,120)( 41,106)
( 42,105)( 43,130)( 44,129)( 45,111)( 46,112)( 47,109)( 48,110)( 49,124)
( 50,123)( 51,116)( 52,115)( 53,102)( 54,101)( 55,103)( 56,104)( 57,121)
( 58,122)( 59,114)( 60,113)( 61,128)( 62,127)( 63,126)( 64,125)( 65,108)
( 66,107)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 74)( 75, 98)( 76, 97)
( 77, 79)( 78, 80)( 81, 92)( 82, 91)( 83, 84)( 93, 96)( 94, 95)(163,292)
(164,291)(165,309)(166,310)(167,312)(168,311)(169,297)(170,298)(171,321)
(172,322)(173,304)(174,303)(175,302)(176,301)(177,315)(178,316)(179,307)
(180,308)(181,293)(182,294)(183,296)(184,295)(185,314)(186,313)(187,305)
(188,306)(189,319)(190,320)(191,317)(192,318)(193,299)(194,300)(195,260)
(196,259)(197,277)(198,278)(199,280)(200,279)(201,265)(202,266)(203,289)
(204,290)(205,272)(206,271)(207,270)(208,269)(209,283)(210,284)(211,275)
(212,276)(213,261)(214,262)(215,264)(216,263)(217,282)(218,281)(219,273)
(220,274)(221,287)(222,288)(223,285)(224,286)(225,267)(226,268)(227,228)
(229,245)(230,246)(231,248)(232,247)(235,257)(236,258)(237,240)(238,239)
(241,251)(242,252)(249,250)(253,255)(254,256);
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2 >;

to this polytope