Polytope of Type {2,8,5}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,5}*640b
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 4
Schlafli Type : {2,8,5}
Number of vertices, edges, etc : 2, 32, 80, 20
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,8,5,2} of size 1280
Vertex Figure Of :
{2,2,8,5} of size 1280
{3,2,8,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,5}*320
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,8,5}*1280a, {2,8,10}*1280b, {2,8,10}*1280d
3-fold covers : {2,8,15}*1920b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)(224,226)
(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)(240,242)
(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)(256,258)
(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)(272,274)
(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)(288,290)
(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)(304,306)
(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)(320,322);;
s2 := ( 3,163)( 4,164)( 5,183)( 6,184)( 7,189)( 8,190)( 9,177)( 10,178)
( 11,172)( 12,171)( 13,191)( 14,192)( 15,182)( 16,181)( 17,169)( 18,170)
( 19,187)( 20,188)( 21,176)( 22,175)( 23,165)( 24,166)( 25,186)( 26,185)
( 27,179)( 28,180)( 29,167)( 30,168)( 31,173)( 32,174)( 33,193)( 34,194)
( 35,291)( 36,292)( 37,311)( 38,312)( 39,317)( 40,318)( 41,305)( 42,306)
( 43,300)( 44,299)( 45,319)( 46,320)( 47,310)( 48,309)( 49,297)( 50,298)
( 51,315)( 52,316)( 53,304)( 54,303)( 55,293)( 56,294)( 57,314)( 58,313)
( 59,307)( 60,308)( 61,295)( 62,296)( 63,301)( 64,302)( 65,321)( 66,322)
( 67,259)( 68,260)( 69,279)( 70,280)( 71,285)( 72,286)( 73,273)( 74,274)
( 75,268)( 76,267)( 77,287)( 78,288)( 79,278)( 80,277)( 81,265)( 82,266)
( 83,283)( 84,284)( 85,272)( 86,271)( 87,261)( 88,262)( 89,282)( 90,281)
( 91,275)( 92,276)( 93,263)( 94,264)( 95,269)( 96,270)( 97,289)( 98,290)
( 99,227)(100,228)(101,247)(102,248)(103,253)(104,254)(105,241)(106,242)
(107,236)(108,235)(109,255)(110,256)(111,246)(112,245)(113,233)(114,234)
(115,251)(116,252)(117,240)(118,239)(119,229)(120,230)(121,250)(122,249)
(123,243)(124,244)(125,231)(126,232)(127,237)(128,238)(129,257)(130,258)
(131,195)(132,196)(133,215)(134,216)(135,221)(136,222)(137,209)(138,210)
(139,204)(140,203)(141,223)(142,224)(143,214)(144,213)(145,201)(146,202)
(147,219)(148,220)(149,208)(150,207)(151,197)(152,198)(153,218)(154,217)
(155,211)(156,212)(157,199)(158,200)(159,205)(160,206)(161,225)(162,226);;
s3 := ( 3,240)( 4,239)( 5,242)( 6,241)( 7,237)( 8,238)( 9,235)( 10,236)
( 11,233)( 12,234)( 13,231)( 14,232)( 15,228)( 16,227)( 17,230)( 18,229)
( 19,246)( 20,245)( 21,244)( 22,243)( 23,248)( 24,247)( 25,250)( 26,249)
( 27,251)( 28,252)( 29,253)( 30,254)( 31,257)( 32,258)( 33,255)( 34,256)
( 35,208)( 36,207)( 37,210)( 38,209)( 39,205)( 40,206)( 41,203)( 42,204)
( 43,201)( 44,202)( 45,199)( 46,200)( 47,196)( 48,195)( 49,198)( 50,197)
( 51,214)( 52,213)( 53,212)( 54,211)( 55,216)( 56,215)( 57,218)( 58,217)
( 59,219)( 60,220)( 61,221)( 62,222)( 63,225)( 64,226)( 65,223)( 66,224)
( 67,176)( 68,175)( 69,178)( 70,177)( 71,173)( 72,174)( 73,171)( 74,172)
( 75,169)( 76,170)( 77,167)( 78,168)( 79,164)( 80,163)( 81,166)( 82,165)
( 83,182)( 84,181)( 85,180)( 86,179)( 87,184)( 88,183)( 89,186)( 90,185)
( 91,187)( 92,188)( 93,189)( 94,190)( 95,193)( 96,194)( 97,191)( 98,192)
( 99,304)(100,303)(101,306)(102,305)(103,301)(104,302)(105,299)(106,300)
(107,297)(108,298)(109,295)(110,296)(111,292)(112,291)(113,294)(114,293)
(115,310)(116,309)(117,308)(118,307)(119,312)(120,311)(121,314)(122,313)
(123,315)(124,316)(125,317)(126,318)(127,321)(128,322)(129,319)(130,320)
(131,272)(132,271)(133,274)(134,273)(135,269)(136,270)(137,267)(138,268)
(139,265)(140,266)(141,263)(142,264)(143,260)(144,259)(145,262)(146,261)
(147,278)(148,277)(149,276)(150,275)(151,280)(152,279)(153,282)(154,281)
(155,283)(156,284)(157,285)(158,286)(159,289)(160,290)(161,287)(162,288);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(322)!(1,2);
s1 := Sym(322)!( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)
(224,226)(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)
(240,242)(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)
(256,258)(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)
(272,274)(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)
(288,290)(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)
(304,306)(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)
(320,322);
s2 := Sym(322)!( 3,163)( 4,164)( 5,183)( 6,184)( 7,189)( 8,190)( 9,177)
( 10,178)( 11,172)( 12,171)( 13,191)( 14,192)( 15,182)( 16,181)( 17,169)
( 18,170)( 19,187)( 20,188)( 21,176)( 22,175)( 23,165)( 24,166)( 25,186)
( 26,185)( 27,179)( 28,180)( 29,167)( 30,168)( 31,173)( 32,174)( 33,193)
( 34,194)( 35,291)( 36,292)( 37,311)( 38,312)( 39,317)( 40,318)( 41,305)
( 42,306)( 43,300)( 44,299)( 45,319)( 46,320)( 47,310)( 48,309)( 49,297)
( 50,298)( 51,315)( 52,316)( 53,304)( 54,303)( 55,293)( 56,294)( 57,314)
( 58,313)( 59,307)( 60,308)( 61,295)( 62,296)( 63,301)( 64,302)( 65,321)
( 66,322)( 67,259)( 68,260)( 69,279)( 70,280)( 71,285)( 72,286)( 73,273)
( 74,274)( 75,268)( 76,267)( 77,287)( 78,288)( 79,278)( 80,277)( 81,265)
( 82,266)( 83,283)( 84,284)( 85,272)( 86,271)( 87,261)( 88,262)( 89,282)
( 90,281)( 91,275)( 92,276)( 93,263)( 94,264)( 95,269)( 96,270)( 97,289)
( 98,290)( 99,227)(100,228)(101,247)(102,248)(103,253)(104,254)(105,241)
(106,242)(107,236)(108,235)(109,255)(110,256)(111,246)(112,245)(113,233)
(114,234)(115,251)(116,252)(117,240)(118,239)(119,229)(120,230)(121,250)
(122,249)(123,243)(124,244)(125,231)(126,232)(127,237)(128,238)(129,257)
(130,258)(131,195)(132,196)(133,215)(134,216)(135,221)(136,222)(137,209)
(138,210)(139,204)(140,203)(141,223)(142,224)(143,214)(144,213)(145,201)
(146,202)(147,219)(148,220)(149,208)(150,207)(151,197)(152,198)(153,218)
(154,217)(155,211)(156,212)(157,199)(158,200)(159,205)(160,206)(161,225)
(162,226);
s3 := Sym(322)!( 3,240)( 4,239)( 5,242)( 6,241)( 7,237)( 8,238)( 9,235)
( 10,236)( 11,233)( 12,234)( 13,231)( 14,232)( 15,228)( 16,227)( 17,230)
( 18,229)( 19,246)( 20,245)( 21,244)( 22,243)( 23,248)( 24,247)( 25,250)
( 26,249)( 27,251)( 28,252)( 29,253)( 30,254)( 31,257)( 32,258)( 33,255)
( 34,256)( 35,208)( 36,207)( 37,210)( 38,209)( 39,205)( 40,206)( 41,203)
( 42,204)( 43,201)( 44,202)( 45,199)( 46,200)( 47,196)( 48,195)( 49,198)
( 50,197)( 51,214)( 52,213)( 53,212)( 54,211)( 55,216)( 56,215)( 57,218)
( 58,217)( 59,219)( 60,220)( 61,221)( 62,222)( 63,225)( 64,226)( 65,223)
( 66,224)( 67,176)( 68,175)( 69,178)( 70,177)( 71,173)( 72,174)( 73,171)
( 74,172)( 75,169)( 76,170)( 77,167)( 78,168)( 79,164)( 80,163)( 81,166)
( 82,165)( 83,182)( 84,181)( 85,180)( 86,179)( 87,184)( 88,183)( 89,186)
( 90,185)( 91,187)( 92,188)( 93,189)( 94,190)( 95,193)( 96,194)( 97,191)
( 98,192)( 99,304)(100,303)(101,306)(102,305)(103,301)(104,302)(105,299)
(106,300)(107,297)(108,298)(109,295)(110,296)(111,292)(112,291)(113,294)
(114,293)(115,310)(116,309)(117,308)(118,307)(119,312)(120,311)(121,314)
(122,313)(123,315)(124,316)(125,317)(126,318)(127,321)(128,322)(129,319)
(130,320)(131,272)(132,271)(133,274)(134,273)(135,269)(136,270)(137,267)
(138,268)(139,265)(140,266)(141,263)(142,264)(143,260)(144,259)(145,262)
(146,261)(147,278)(148,277)(149,276)(150,275)(151,280)(152,279)(153,282)
(154,281)(155,283)(156,284)(157,285)(158,286)(159,289)(160,290)(161,287)
(162,288);
poly := sub<Sym(322)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 >;
to this polytope