Polytope of Type {8,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,10}*640d
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 3
Schlafli Type : {8,10}
Number of vertices, edges, etc : 32, 160, 40
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 :
   {8,10,2} of size 1280
Vertex Figure Of :
   {2,8,10} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,5}*320b, {4,10}*320b
   4-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,20}*1280g, {8,20}*1280h, {8,10}*1280c
   3-fold covers : {24,10}*1920a, {8,30}*1920k
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

s0 := Sym(320)!(  1,163)(  2,164)(  3,161)(  4,162)(  5,167)(  6,168)(  7,165)
(  8,166)(  9,171)( 10,172)( 11,169)( 12,170)( 13,175)( 14,176)( 15,173)
( 16,174)( 17,179)( 18,180)( 19,177)( 20,178)( 21,183)( 22,184)( 23,181)
( 24,182)( 25,187)( 26,188)( 27,185)( 28,186)( 29,191)( 30,192)( 31,189)
( 32,190)( 33,195)( 34,196)( 35,193)( 36,194)( 37,199)( 38,200)( 39,197)
( 40,198)( 41,203)( 42,204)( 43,201)( 44,202)( 45,207)( 46,208)( 47,205)
( 48,206)( 49,211)( 50,212)( 51,209)( 52,210)( 53,215)( 54,216)( 55,213)
( 56,214)( 57,219)( 58,220)( 59,217)( 60,218)( 61,223)( 62,224)( 63,221)
( 64,222)( 65,227)( 66,228)( 67,225)( 68,226)( 69,231)( 70,232)( 71,229)
( 72,230)( 73,235)( 74,236)( 75,233)( 76,234)( 77,239)( 78,240)( 79,237)
( 80,238)( 81,243)( 82,244)( 83,241)( 84,242)( 85,247)( 86,248)( 87,245)
( 88,246)( 89,251)( 90,252)( 91,249)( 92,250)( 93,255)( 94,256)( 95,253)
( 96,254)( 97,259)( 98,260)( 99,257)(100,258)(101,263)(102,264)(103,261)
(104,262)(105,267)(106,268)(107,265)(108,266)(109,271)(110,272)(111,269)
(112,270)(113,275)(114,276)(115,273)(116,274)(117,279)(118,280)(119,277)
(120,278)(121,283)(122,284)(123,281)(124,282)(125,287)(126,288)(127,285)
(128,286)(129,291)(130,292)(131,289)(132,290)(133,295)(134,296)(135,293)
(136,294)(137,299)(138,300)(139,297)(140,298)(141,303)(142,304)(143,301)
(144,302)(145,307)(146,308)(147,305)(148,306)(149,311)(150,312)(151,309)
(152,310)(153,315)(154,316)(155,313)(156,314)(157,319)(158,320)(159,317)
(160,318);
s1 := 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);
s2 := Sym(320)!(  1,259)(  2,260)(  3,257)(  4,258)(  5,262)(  6,261)(  7,264)
(  8,263)(  9,265)( 10,266)( 11,267)( 12,268)( 13,272)( 14,271)( 15,270)
( 16,269)( 17,286)( 18,285)( 19,288)( 20,287)( 21,284)( 22,283)( 23,282)
( 24,281)( 25,280)( 26,279)( 27,278)( 28,277)( 29,274)( 30,273)( 31,276)
( 32,275)( 33,227)( 34,228)( 35,225)( 36,226)( 37,230)( 38,229)( 39,232)
( 40,231)( 41,233)( 42,234)( 43,235)( 44,236)( 45,240)( 46,239)( 47,238)
( 48,237)( 49,254)( 50,253)( 51,256)( 52,255)( 53,252)( 54,251)( 55,250)
( 56,249)( 57,248)( 58,247)( 59,246)( 60,245)( 61,242)( 62,241)( 63,244)
( 64,243)( 65,195)( 66,196)( 67,193)( 68,194)( 69,198)( 70,197)( 71,200)
( 72,199)( 73,201)( 74,202)( 75,203)( 76,204)( 77,208)( 78,207)( 79,206)
( 80,205)( 81,222)( 82,221)( 83,224)( 84,223)( 85,220)( 86,219)( 87,218)
( 88,217)( 89,216)( 90,215)( 91,214)( 92,213)( 93,210)( 94,209)( 95,212)
( 96,211)( 97,163)( 98,164)( 99,161)(100,162)(101,166)(102,165)(103,168)
(104,167)(105,169)(106,170)(107,171)(108,172)(109,176)(110,175)(111,174)
(112,173)(113,190)(114,189)(115,192)(116,191)(117,188)(118,187)(119,186)
(120,185)(121,184)(122,183)(123,182)(124,181)(125,178)(126,177)(127,180)
(128,179)(129,291)(130,292)(131,289)(132,290)(133,294)(134,293)(135,296)
(136,295)(137,297)(138,298)(139,299)(140,300)(141,304)(142,303)(143,302)
(144,301)(145,318)(146,317)(147,320)(148,319)(149,316)(150,315)(151,314)
(152,313)(153,312)(154,311)(155,310)(156,309)(157,306)(158,305)(159,308)
(160,307);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0 >;

References : None.
to this polytope