Polytope of Type {8,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,5}*640a
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 3
Schlafli Type : {8,5}
Number of vertices, edges, etc : 64, 160, 40
Order of s₀s₁s₂ : 10
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,5,2} of size 1280
Vertex Figure Of :
   {2,8,5} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,5}*320a, {8,5}*320b, {4,5}*320
   4-fold quotients : {4,5}*160
   32-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,5}*1280, {8,10}*1280a, {8,10}*1280c
   3-fold covers : {8,15}*1920b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope