Polytope of Type {8,5}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,5}*640b
if this polytope has a name.
Group : SmallGroup(640,21465)
Rank : 3
Schlafli Type : {8,5}
Number of vertices, edges, etc : 64, 160, 40
Order of s0s1s2 : 20
Order of s0s1s2s1 : 4
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 : {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}*1280b, {8,10}*1280d
3-fold covers : {8,15}*1920c
Permutation Representation (GAP) :
s0 := ( 1,177)( 2,178)( 3,180)( 4,179)( 5,182)( 6,181)( 7,183)( 8,184)
( 9,186)( 10,185)( 11,187)( 12,188)( 13,189)( 14,190)( 15,192)( 16,191)
( 17,162)( 18,161)( 19,163)( 20,164)( 21,165)( 22,166)( 23,168)( 24,167)
( 25,169)( 26,170)( 27,172)( 28,171)( 29,174)( 30,173)( 31,175)( 32,176)
( 33,209)( 34,210)( 35,212)( 36,211)( 37,214)( 38,213)( 39,215)( 40,216)
( 41,218)( 42,217)( 43,219)( 44,220)( 45,221)( 46,222)( 47,224)( 48,223)
( 49,194)( 50,193)( 51,195)( 52,196)( 53,197)( 54,198)( 55,200)( 56,199)
( 57,201)( 58,202)( 59,204)( 60,203)( 61,206)( 62,205)( 63,207)( 64,208)
( 65,241)( 66,242)( 67,244)( 68,243)( 69,246)( 70,245)( 71,247)( 72,248)
( 73,250)( 74,249)( 75,251)( 76,252)( 77,253)( 78,254)( 79,256)( 80,255)
( 81,226)( 82,225)( 83,227)( 84,228)( 85,229)( 86,230)( 87,232)( 88,231)
( 89,233)( 90,234)( 91,236)( 92,235)( 93,238)( 94,237)( 95,239)( 96,240)
( 97,273)( 98,274)( 99,276)(100,275)(101,278)(102,277)(103,279)(104,280)
(105,282)(106,281)(107,283)(108,284)(109,285)(110,286)(111,288)(112,287)
(113,258)(114,257)(115,259)(116,260)(117,261)(118,262)(119,264)(120,263)
(121,265)(122,266)(123,268)(124,267)(125,270)(126,269)(127,271)(128,272)
(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)(136,312)
(137,314)(138,313)(139,315)(140,316)(141,317)(142,318)(143,320)(144,319)
(145,290)(146,289)(147,291)(148,292)(149,293)(150,294)(151,296)(152,295)
(153,297)(154,298)(155,300)(156,299)(157,302)(158,301)(159,303)(160,304);;
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)(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);;
s2 := ( 1,129)( 2,130)( 3,148)( 4,147)( 5,149)( 6,150)( 7,136)( 8,135)
( 9,160)( 10,159)( 11,141)( 12,142)( 13,139)( 14,140)( 15,154)( 16,153)
( 17,146)( 18,145)( 19,132)( 20,131)( 21,133)( 22,134)( 23,151)( 24,152)
( 25,144)( 26,143)( 27,158)( 28,157)( 29,156)( 30,155)( 31,138)( 32,137)
( 33, 97)( 34, 98)( 35,116)( 36,115)( 37,117)( 38,118)( 39,104)( 40,103)
( 41,128)( 42,127)( 43,109)( 44,110)( 45,107)( 46,108)( 47,122)( 48,121)
( 49,114)( 50,113)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,119)( 56,120)
( 57,112)( 58,111)( 59,126)( 60,125)( 61,124)( 62,123)( 63,106)( 64,105)
( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 72)( 73, 96)( 74, 95)( 75, 77)
( 76, 78)( 79, 90)( 80, 89)( 81, 82)( 91, 94)( 92, 93)(161,290)(162,289)
(163,307)(164,308)(165,310)(166,309)(167,295)(168,296)(169,319)(170,320)
(171,302)(172,301)(173,300)(174,299)(175,313)(176,314)(177,305)(178,306)
(179,291)(180,292)(181,294)(182,293)(183,312)(184,311)(185,303)(186,304)
(187,317)(188,318)(189,315)(190,316)(191,297)(192,298)(193,258)(194,257)
(195,275)(196,276)(197,278)(198,277)(199,263)(200,264)(201,287)(202,288)
(203,270)(204,269)(205,268)(206,267)(207,281)(208,282)(209,273)(210,274)
(211,259)(212,260)(213,262)(214,261)(215,280)(216,279)(217,271)(218,272)
(219,285)(220,286)(221,283)(222,284)(223,265)(224,266)(225,226)(227,243)
(228,244)(229,246)(230,245)(233,255)(234,256)(235,238)(236,237)(239,249)
(240,250)(247,248)(251,253)(252,254);;
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,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(320)!( 1,177)( 2,178)( 3,180)( 4,179)( 5,182)( 6,181)( 7,183)
( 8,184)( 9,186)( 10,185)( 11,187)( 12,188)( 13,189)( 14,190)( 15,192)
( 16,191)( 17,162)( 18,161)( 19,163)( 20,164)( 21,165)( 22,166)( 23,168)
( 24,167)( 25,169)( 26,170)( 27,172)( 28,171)( 29,174)( 30,173)( 31,175)
( 32,176)( 33,209)( 34,210)( 35,212)( 36,211)( 37,214)( 38,213)( 39,215)
( 40,216)( 41,218)( 42,217)( 43,219)( 44,220)( 45,221)( 46,222)( 47,224)
( 48,223)( 49,194)( 50,193)( 51,195)( 52,196)( 53,197)( 54,198)( 55,200)
( 56,199)( 57,201)( 58,202)( 59,204)( 60,203)( 61,206)( 62,205)( 63,207)
( 64,208)( 65,241)( 66,242)( 67,244)( 68,243)( 69,246)( 70,245)( 71,247)
( 72,248)( 73,250)( 74,249)( 75,251)( 76,252)( 77,253)( 78,254)( 79,256)
( 80,255)( 81,226)( 82,225)( 83,227)( 84,228)( 85,229)( 86,230)( 87,232)
( 88,231)( 89,233)( 90,234)( 91,236)( 92,235)( 93,238)( 94,237)( 95,239)
( 96,240)( 97,273)( 98,274)( 99,276)(100,275)(101,278)(102,277)(103,279)
(104,280)(105,282)(106,281)(107,283)(108,284)(109,285)(110,286)(111,288)
(112,287)(113,258)(114,257)(115,259)(116,260)(117,261)(118,262)(119,264)
(120,263)(121,265)(122,266)(123,268)(124,267)(125,270)(126,269)(127,271)
(128,272)(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)
(136,312)(137,314)(138,313)(139,315)(140,316)(141,317)(142,318)(143,320)
(144,319)(145,290)(146,289)(147,291)(148,292)(149,293)(150,294)(151,296)
(152,295)(153,297)(154,298)(155,300)(156,299)(157,302)(158,301)(159,303)
(160,304);
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)(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);
s2 := Sym(320)!( 1,129)( 2,130)( 3,148)( 4,147)( 5,149)( 6,150)( 7,136)
( 8,135)( 9,160)( 10,159)( 11,141)( 12,142)( 13,139)( 14,140)( 15,154)
( 16,153)( 17,146)( 18,145)( 19,132)( 20,131)( 21,133)( 22,134)( 23,151)
( 24,152)( 25,144)( 26,143)( 27,158)( 28,157)( 29,156)( 30,155)( 31,138)
( 32,137)( 33, 97)( 34, 98)( 35,116)( 36,115)( 37,117)( 38,118)( 39,104)
( 40,103)( 41,128)( 42,127)( 43,109)( 44,110)( 45,107)( 46,108)( 47,122)
( 48,121)( 49,114)( 50,113)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,119)
( 56,120)( 57,112)( 58,111)( 59,126)( 60,125)( 61,124)( 62,123)( 63,106)
( 64,105)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 72)( 73, 96)( 74, 95)
( 75, 77)( 76, 78)( 79, 90)( 80, 89)( 81, 82)( 91, 94)( 92, 93)(161,290)
(162,289)(163,307)(164,308)(165,310)(166,309)(167,295)(168,296)(169,319)
(170,320)(171,302)(172,301)(173,300)(174,299)(175,313)(176,314)(177,305)
(178,306)(179,291)(180,292)(181,294)(182,293)(183,312)(184,311)(185,303)
(186,304)(187,317)(188,318)(189,315)(190,316)(191,297)(192,298)(193,258)
(194,257)(195,275)(196,276)(197,278)(198,277)(199,263)(200,264)(201,287)
(202,288)(203,270)(204,269)(205,268)(206,267)(207,281)(208,282)(209,273)
(210,274)(211,259)(212,260)(213,262)(214,261)(215,280)(216,279)(217,271)
(218,272)(219,285)(220,286)(221,283)(222,284)(223,265)(224,266)(225,226)
(227,243)(228,244)(229,246)(230,245)(233,255)(234,256)(235,238)(236,237)
(239,249)(240,250)(247,248)(251,253)(252,254);
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,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope