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