Polytope of Type {5,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,4}*640
if this polytope has a name.
Group : SmallGroup(640,21465)
Rank : 3
Schlafli Type : {5,4}
Number of vertices, edges, etc : 80, 160, 64
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {5,4,2} of size 1280
Vertex Figure Of :
   {2,5,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {5,4}*320
   4-fold quotients : {5,4}*160
   32-fold quotients : {5,2}*20
Covers (Minimal Covers in Boldface) :
   2-fold covers : {5,8}*1280, {10,4}*1280b
   3-fold covers : {15,4}*1920a
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope