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