Polytope of Type {10,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,8}*640a
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 3
Schlafli Type : {10,8}
Number of vertices, edges, etc : 40, 160, 32
Order of s0s1s2 : 10
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{10,8,2} of size 1280
Vertex Figure Of :
{2,10,8} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,8}*320b, {10,4}*320a
4-fold quotients : {5,4}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,8}*1280i, {20,8}*1280k, {10,8}*1280c
3-fold covers : {30,8}*1920h
Permutation Representation (GAP) :
s0 := ( 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)(163,188)(164,187)
(165,175)(166,176)(167,181)(168,182)(169,178)(170,177)(171,172)(173,192)
(174,191)(183,189)(184,190)(185,186)(193,289)(194,290)(195,316)(196,315)
(197,303)(198,304)(199,309)(200,310)(201,306)(202,305)(203,300)(204,299)
(205,320)(206,319)(207,293)(208,294)(209,298)(210,297)(211,307)(212,308)
(213,295)(214,296)(215,317)(216,318)(217,314)(218,313)(219,292)(220,291)
(221,311)(222,312)(223,302)(224,301)(225,257)(226,258)(227,284)(228,283)
(229,271)(230,272)(231,277)(232,278)(233,274)(234,273)(235,268)(236,267)
(237,288)(238,287)(239,261)(240,262)(241,266)(242,265)(243,275)(244,276)
(245,263)(246,264)(247,285)(248,286)(249,282)(250,281)(251,260)(252,259)
(253,279)(254,280)(255,270)(256,269);;
s1 := ( 1,193)( 2,194)( 3,215)( 4,216)( 5,217)( 6,218)( 7,207)( 8,208)
( 9,202)( 10,201)( 11,223)( 12,224)( 13,210)( 14,209)( 15,199)( 16,200)
( 17,206)( 18,205)( 19,219)( 20,220)( 21,214)( 22,213)( 23,195)( 24,196)
( 25,197)( 26,198)( 27,211)( 28,212)( 29,221)( 30,222)( 31,203)( 32,204)
( 33,161)( 34,162)( 35,183)( 36,184)( 37,185)( 38,186)( 39,175)( 40,176)
( 41,170)( 42,169)( 43,191)( 44,192)( 45,178)( 46,177)( 47,167)( 48,168)
( 49,174)( 50,173)( 51,187)( 52,188)( 53,182)( 54,181)( 55,163)( 56,164)
( 57,165)( 58,166)( 59,179)( 60,180)( 61,189)( 62,190)( 63,171)( 64,172)
( 65,289)( 66,290)( 67,311)( 68,312)( 69,313)( 70,314)( 71,303)( 72,304)
( 73,298)( 74,297)( 75,319)( 76,320)( 77,306)( 78,305)( 79,295)( 80,296)
( 81,302)( 82,301)( 83,315)( 84,316)( 85,310)( 86,309)( 87,291)( 88,292)
( 89,293)( 90,294)( 91,307)( 92,308)( 93,317)( 94,318)( 95,299)( 96,300)
( 97,257)( 98,258)( 99,279)(100,280)(101,281)(102,282)(103,271)(104,272)
(105,266)(106,265)(107,287)(108,288)(109,274)(110,273)(111,263)(112,264)
(113,270)(114,269)(115,283)(116,284)(117,278)(118,277)(119,259)(120,260)
(121,261)(122,262)(123,275)(124,276)(125,285)(126,286)(127,267)(128,268)
(129,225)(130,226)(131,247)(132,248)(133,249)(134,250)(135,239)(136,240)
(137,234)(138,233)(139,255)(140,256)(141,242)(142,241)(143,231)(144,232)
(145,238)(146,237)(147,251)(148,252)(149,246)(150,245)(151,227)(152,228)
(153,229)(154,230)(155,243)(156,244)(157,253)(158,254)(159,235)(160,236);;
s2 := ( 1, 19)( 2, 20)( 3, 18)( 4, 17)( 5, 24)( 6, 23)( 7, 21)( 8, 22)
( 9, 28)( 10, 27)( 11, 25)( 12, 26)( 13, 31)( 14, 32)( 15, 30)( 16, 29)
( 33, 51)( 34, 52)( 35, 50)( 36, 49)( 37, 56)( 38, 55)( 39, 53)( 40, 54)
( 41, 60)( 42, 59)( 43, 57)( 44, 58)( 45, 63)( 46, 64)( 47, 62)( 48, 61)
( 65, 83)( 66, 84)( 67, 82)( 68, 81)( 69, 88)( 70, 87)( 71, 85)( 72, 86)
( 73, 92)( 74, 91)( 75, 89)( 76, 90)( 77, 95)( 78, 96)( 79, 94)( 80, 93)
( 97,115)( 98,116)( 99,114)(100,113)(101,120)(102,119)(103,117)(104,118)
(105,124)(106,123)(107,121)(108,122)(109,127)(110,128)(111,126)(112,125)
(129,147)(130,148)(131,146)(132,145)(133,152)(134,151)(135,149)(136,150)
(137,156)(138,155)(139,153)(140,154)(141,159)(142,160)(143,158)(144,157)
(161,179)(162,180)(163,178)(164,177)(165,184)(166,183)(167,181)(168,182)
(169,188)(170,187)(171,185)(172,186)(173,191)(174,192)(175,190)(176,189)
(193,211)(194,212)(195,210)(196,209)(197,216)(198,215)(199,213)(200,214)
(201,220)(202,219)(203,217)(204,218)(205,223)(206,224)(207,222)(208,221)
(225,243)(226,244)(227,242)(228,241)(229,248)(230,247)(231,245)(232,246)
(233,252)(234,251)(235,249)(236,250)(237,255)(238,256)(239,254)(240,253)
(257,275)(258,276)(259,274)(260,273)(261,280)(262,279)(263,277)(264,278)
(265,284)(266,283)(267,281)(268,282)(269,287)(270,288)(271,286)(272,285)
(289,307)(290,308)(291,306)(292,305)(293,312)(294,311)(295,309)(296,310)
(297,316)(298,315)(299,313)(300,314)(301,319)(302,320)(303,318)(304,317);;
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s2,
s2*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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)(163,188)
(164,187)(165,175)(166,176)(167,181)(168,182)(169,178)(170,177)(171,172)
(173,192)(174,191)(183,189)(184,190)(185,186)(193,289)(194,290)(195,316)
(196,315)(197,303)(198,304)(199,309)(200,310)(201,306)(202,305)(203,300)
(204,299)(205,320)(206,319)(207,293)(208,294)(209,298)(210,297)(211,307)
(212,308)(213,295)(214,296)(215,317)(216,318)(217,314)(218,313)(219,292)
(220,291)(221,311)(222,312)(223,302)(224,301)(225,257)(226,258)(227,284)
(228,283)(229,271)(230,272)(231,277)(232,278)(233,274)(234,273)(235,268)
(236,267)(237,288)(238,287)(239,261)(240,262)(241,266)(242,265)(243,275)
(244,276)(245,263)(246,264)(247,285)(248,286)(249,282)(250,281)(251,260)
(252,259)(253,279)(254,280)(255,270)(256,269);
s1 := Sym(320)!( 1,193)( 2,194)( 3,215)( 4,216)( 5,217)( 6,218)( 7,207)
( 8,208)( 9,202)( 10,201)( 11,223)( 12,224)( 13,210)( 14,209)( 15,199)
( 16,200)( 17,206)( 18,205)( 19,219)( 20,220)( 21,214)( 22,213)( 23,195)
( 24,196)( 25,197)( 26,198)( 27,211)( 28,212)( 29,221)( 30,222)( 31,203)
( 32,204)( 33,161)( 34,162)( 35,183)( 36,184)( 37,185)( 38,186)( 39,175)
( 40,176)( 41,170)( 42,169)( 43,191)( 44,192)( 45,178)( 46,177)( 47,167)
( 48,168)( 49,174)( 50,173)( 51,187)( 52,188)( 53,182)( 54,181)( 55,163)
( 56,164)( 57,165)( 58,166)( 59,179)( 60,180)( 61,189)( 62,190)( 63,171)
( 64,172)( 65,289)( 66,290)( 67,311)( 68,312)( 69,313)( 70,314)( 71,303)
( 72,304)( 73,298)( 74,297)( 75,319)( 76,320)( 77,306)( 78,305)( 79,295)
( 80,296)( 81,302)( 82,301)( 83,315)( 84,316)( 85,310)( 86,309)( 87,291)
( 88,292)( 89,293)( 90,294)( 91,307)( 92,308)( 93,317)( 94,318)( 95,299)
( 96,300)( 97,257)( 98,258)( 99,279)(100,280)(101,281)(102,282)(103,271)
(104,272)(105,266)(106,265)(107,287)(108,288)(109,274)(110,273)(111,263)
(112,264)(113,270)(114,269)(115,283)(116,284)(117,278)(118,277)(119,259)
(120,260)(121,261)(122,262)(123,275)(124,276)(125,285)(126,286)(127,267)
(128,268)(129,225)(130,226)(131,247)(132,248)(133,249)(134,250)(135,239)
(136,240)(137,234)(138,233)(139,255)(140,256)(141,242)(142,241)(143,231)
(144,232)(145,238)(146,237)(147,251)(148,252)(149,246)(150,245)(151,227)
(152,228)(153,229)(154,230)(155,243)(156,244)(157,253)(158,254)(159,235)
(160,236);
s2 := Sym(320)!( 1, 19)( 2, 20)( 3, 18)( 4, 17)( 5, 24)( 6, 23)( 7, 21)
( 8, 22)( 9, 28)( 10, 27)( 11, 25)( 12, 26)( 13, 31)( 14, 32)( 15, 30)
( 16, 29)( 33, 51)( 34, 52)( 35, 50)( 36, 49)( 37, 56)( 38, 55)( 39, 53)
( 40, 54)( 41, 60)( 42, 59)( 43, 57)( 44, 58)( 45, 63)( 46, 64)( 47, 62)
( 48, 61)( 65, 83)( 66, 84)( 67, 82)( 68, 81)( 69, 88)( 70, 87)( 71, 85)
( 72, 86)( 73, 92)( 74, 91)( 75, 89)( 76, 90)( 77, 95)( 78, 96)( 79, 94)
( 80, 93)( 97,115)( 98,116)( 99,114)(100,113)(101,120)(102,119)(103,117)
(104,118)(105,124)(106,123)(107,121)(108,122)(109,127)(110,128)(111,126)
(112,125)(129,147)(130,148)(131,146)(132,145)(133,152)(134,151)(135,149)
(136,150)(137,156)(138,155)(139,153)(140,154)(141,159)(142,160)(143,158)
(144,157)(161,179)(162,180)(163,178)(164,177)(165,184)(166,183)(167,181)
(168,182)(169,188)(170,187)(171,185)(172,186)(173,191)(174,192)(175,190)
(176,189)(193,211)(194,212)(195,210)(196,209)(197,216)(198,215)(199,213)
(200,214)(201,220)(202,219)(203,217)(204,218)(205,223)(206,224)(207,222)
(208,221)(225,243)(226,244)(227,242)(228,241)(229,248)(230,247)(231,245)
(232,246)(233,252)(234,251)(235,249)(236,250)(237,255)(238,256)(239,254)
(240,253)(257,275)(258,276)(259,274)(260,273)(261,280)(262,279)(263,277)
(264,278)(265,284)(266,283)(267,281)(268,282)(269,287)(270,288)(271,286)
(272,285)(289,307)(290,308)(291,306)(292,305)(293,312)(294,311)(295,309)
(296,310)(297,316)(298,315)(299,313)(300,314)(301,319)(302,320)(303,318)
(304,317);
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s2,
s2*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope