Polytope of Type {2,5,4}

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

to this polytope