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Polytope of Type {8,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,5}*320b
if this polytope has a name.
Group : SmallGroup(320,1582)
Rank : 3
Schlafli Type : {8,5}
Number of vertices, edges, etc : 32, 80, 20
Order of s0s1s2 : 5
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,5,2} of size 640
Vertex Figure Of :
   {2,8,5} of size 640
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,5}*640a, {8,10}*640b, {8,10}*640d
   3-fold covers : {8,15}*960b
   4-fold covers : {8,5}*1280, {8,10}*1280a, {8,20}*1280g, {8,20}*1280h, {8,20}*1280j, {8,20}*1280l, {8,10}*1280c
   5-fold covers : {8,25}*1600b
   6-fold covers : {24,10}*1920a, {8,15}*1920b, {8,30}*1920i, {8,30}*1920k
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320);;
s1 := (  1,161)(  2,162)(  3,181)(  4,182)(  5,187)(  6,188)(  7,175)(  8,176)
(  9,170)( 10,169)( 11,189)( 12,190)( 13,180)( 14,179)( 15,167)( 16,168)
( 17,185)( 18,186)( 19,174)( 20,173)( 21,163)( 22,164)( 23,184)( 24,183)
( 25,177)( 26,178)( 27,165)( 28,166)( 29,171)( 30,172)( 31,191)( 32,192)
( 33,289)( 34,290)( 35,309)( 36,310)( 37,315)( 38,316)( 39,303)( 40,304)
( 41,298)( 42,297)( 43,317)( 44,318)( 45,308)( 46,307)( 47,295)( 48,296)
( 49,313)( 50,314)( 51,302)( 52,301)( 53,291)( 54,292)( 55,312)( 56,311)
( 57,305)( 58,306)( 59,293)( 60,294)( 61,299)( 62,300)( 63,319)( 64,320)
( 65,257)( 66,258)( 67,277)( 68,278)( 69,283)( 70,284)( 71,271)( 72,272)
( 73,266)( 74,265)( 75,285)( 76,286)( 77,276)( 78,275)( 79,263)( 80,264)
( 81,281)( 82,282)( 83,270)( 84,269)( 85,259)( 86,260)( 87,280)( 88,279)
( 89,273)( 90,274)( 91,261)( 92,262)( 93,267)( 94,268)( 95,287)( 96,288)
( 97,225)( 98,226)( 99,245)(100,246)(101,251)(102,252)(103,239)(104,240)
(105,234)(106,233)(107,253)(108,254)(109,244)(110,243)(111,231)(112,232)
(113,249)(114,250)(115,238)(116,237)(117,227)(118,228)(119,248)(120,247)
(121,241)(122,242)(123,229)(124,230)(125,235)(126,236)(127,255)(128,256)
(129,193)(130,194)(131,213)(132,214)(133,219)(134,220)(135,207)(136,208)
(137,202)(138,201)(139,221)(140,222)(141,212)(142,211)(143,199)(144,200)
(145,217)(146,218)(147,206)(148,205)(149,195)(150,196)(151,216)(152,215)
(153,209)(154,210)(155,197)(156,198)(157,203)(158,204)(159,223)(160,224);;
s2 := (  1,238)(  2,237)(  3,240)(  4,239)(  5,235)(  6,236)(  7,233)(  8,234)
(  9,231)( 10,232)( 11,229)( 12,230)( 13,226)( 14,225)( 15,228)( 16,227)
( 17,244)( 18,243)( 19,242)( 20,241)( 21,246)( 22,245)( 23,248)( 24,247)
( 25,249)( 26,250)( 27,251)( 28,252)( 29,255)( 30,256)( 31,253)( 32,254)
( 33,206)( 34,205)( 35,208)( 36,207)( 37,203)( 38,204)( 39,201)( 40,202)
( 41,199)( 42,200)( 43,197)( 44,198)( 45,194)( 46,193)( 47,196)( 48,195)
( 49,212)( 50,211)( 51,210)( 52,209)( 53,214)( 54,213)( 55,216)( 56,215)
( 57,217)( 58,218)( 59,219)( 60,220)( 61,223)( 62,224)( 63,221)( 64,222)
( 65,174)( 66,173)( 67,176)( 68,175)( 69,171)( 70,172)( 71,169)( 72,170)
( 73,167)( 74,168)( 75,165)( 76,166)( 77,162)( 78,161)( 79,164)( 80,163)
( 81,180)( 82,179)( 83,178)( 84,177)( 85,182)( 86,181)( 87,184)( 88,183)
( 89,185)( 90,186)( 91,187)( 92,188)( 93,191)( 94,192)( 95,189)( 96,190)
( 97,302)( 98,301)( 99,304)(100,303)(101,299)(102,300)(103,297)(104,298)
(105,295)(106,296)(107,293)(108,294)(109,290)(110,289)(111,292)(112,291)
(113,308)(114,307)(115,306)(116,305)(117,310)(118,309)(119,312)(120,311)
(121,313)(122,314)(123,315)(124,316)(125,319)(126,320)(127,317)(128,318)
(129,270)(130,269)(131,272)(132,271)(133,267)(134,268)(135,265)(136,266)
(137,263)(138,264)(139,261)(140,262)(141,258)(142,257)(143,260)(144,259)
(145,276)(146,275)(147,274)(148,273)(149,278)(150,277)(151,280)(152,279)
(153,281)(154,282)(155,283)(156,284)(157,287)(158,288)(159,285)(160,286);;
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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(320)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320);
s1 := Sym(320)!(  1,161)(  2,162)(  3,181)(  4,182)(  5,187)(  6,188)(  7,175)
(  8,176)(  9,170)( 10,169)( 11,189)( 12,190)( 13,180)( 14,179)( 15,167)
( 16,168)( 17,185)( 18,186)( 19,174)( 20,173)( 21,163)( 22,164)( 23,184)
( 24,183)( 25,177)( 26,178)( 27,165)( 28,166)( 29,171)( 30,172)( 31,191)
( 32,192)( 33,289)( 34,290)( 35,309)( 36,310)( 37,315)( 38,316)( 39,303)
( 40,304)( 41,298)( 42,297)( 43,317)( 44,318)( 45,308)( 46,307)( 47,295)
( 48,296)( 49,313)( 50,314)( 51,302)( 52,301)( 53,291)( 54,292)( 55,312)
( 56,311)( 57,305)( 58,306)( 59,293)( 60,294)( 61,299)( 62,300)( 63,319)
( 64,320)( 65,257)( 66,258)( 67,277)( 68,278)( 69,283)( 70,284)( 71,271)
( 72,272)( 73,266)( 74,265)( 75,285)( 76,286)( 77,276)( 78,275)( 79,263)
( 80,264)( 81,281)( 82,282)( 83,270)( 84,269)( 85,259)( 86,260)( 87,280)
( 88,279)( 89,273)( 90,274)( 91,261)( 92,262)( 93,267)( 94,268)( 95,287)
( 96,288)( 97,225)( 98,226)( 99,245)(100,246)(101,251)(102,252)(103,239)
(104,240)(105,234)(106,233)(107,253)(108,254)(109,244)(110,243)(111,231)
(112,232)(113,249)(114,250)(115,238)(116,237)(117,227)(118,228)(119,248)
(120,247)(121,241)(122,242)(123,229)(124,230)(125,235)(126,236)(127,255)
(128,256)(129,193)(130,194)(131,213)(132,214)(133,219)(134,220)(135,207)
(136,208)(137,202)(138,201)(139,221)(140,222)(141,212)(142,211)(143,199)
(144,200)(145,217)(146,218)(147,206)(148,205)(149,195)(150,196)(151,216)
(152,215)(153,209)(154,210)(155,197)(156,198)(157,203)(158,204)(159,223)
(160,224);
s2 := Sym(320)!(  1,238)(  2,237)(  3,240)(  4,239)(  5,235)(  6,236)(  7,233)
(  8,234)(  9,231)( 10,232)( 11,229)( 12,230)( 13,226)( 14,225)( 15,228)
( 16,227)( 17,244)( 18,243)( 19,242)( 20,241)( 21,246)( 22,245)( 23,248)
( 24,247)( 25,249)( 26,250)( 27,251)( 28,252)( 29,255)( 30,256)( 31,253)
( 32,254)( 33,206)( 34,205)( 35,208)( 36,207)( 37,203)( 38,204)( 39,201)
( 40,202)( 41,199)( 42,200)( 43,197)( 44,198)( 45,194)( 46,193)( 47,196)
( 48,195)( 49,212)( 50,211)( 51,210)( 52,209)( 53,214)( 54,213)( 55,216)
( 56,215)( 57,217)( 58,218)( 59,219)( 60,220)( 61,223)( 62,224)( 63,221)
( 64,222)( 65,174)( 66,173)( 67,176)( 68,175)( 69,171)( 70,172)( 71,169)
( 72,170)( 73,167)( 74,168)( 75,165)( 76,166)( 77,162)( 78,161)( 79,164)
( 80,163)( 81,180)( 82,179)( 83,178)( 84,177)( 85,182)( 86,181)( 87,184)
( 88,183)( 89,185)( 90,186)( 91,187)( 92,188)( 93,191)( 94,192)( 95,189)
( 96,190)( 97,302)( 98,301)( 99,304)(100,303)(101,299)(102,300)(103,297)
(104,298)(105,295)(106,296)(107,293)(108,294)(109,290)(110,289)(111,292)
(112,291)(113,308)(114,307)(115,306)(116,305)(117,310)(118,309)(119,312)
(120,311)(121,313)(122,314)(123,315)(124,316)(125,319)(126,320)(127,317)
(128,318)(129,270)(130,269)(131,272)(132,271)(133,267)(134,268)(135,265)
(136,266)(137,263)(138,264)(139,261)(140,262)(141,258)(142,257)(143,260)
(144,259)(145,276)(146,275)(147,274)(148,273)(149,278)(150,277)(151,280)
(152,279)(153,281)(154,282)(155,283)(156,284)(157,287)(158,288)(159,285)
(160,286);
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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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 >; 
 
References : None.
to this polytope