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