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