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Polytope of Type {4,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,5}*320
Also Known As : {4,5|4}. if this polytope has another name.
Group : SmallGroup(320,1636)
Rank : 3
Schlafli Type : {4,5}
Number of vertices, edges, etc : 32, 80, 40
Order of s0s1s2 : 10
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,5,2} of size 640
   {4,5,3} of size 1920
   {4,5,5} of size 1920
   {4,5,3} of size 1920
Vertex Figure Of :
   {2,4,5} of size 640
   {4,4,5} of size 1280
   {6,4,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,5}*160
   16-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,5}*640a, {4,5}*640, {4,10}*640a, {8,5}*640b, {4,10}*640b
   3-fold covers : {4,15}*960
   4-fold covers : {8,5}*1280, {8,10}*1280a, {8,10}*1280b, {4,10}*1280a, {4,20}*1280b, {4,20}*1280c, {8,10}*1280c, {4,10}*1280b, {4,20}*1280d, {8,10}*1280d, {4,20}*1280e, {4,10}*1280c, {8,10}*1280e, {8,10}*1280f
   5-fold covers : {4,25}*1600, {20,5}*1600
   6-fold covers : {8,15}*1920b, {4,15}*1920a, {4,30}*1920c, {8,15}*1920c, {12,10}*1920a, {4,30}*1920d
Permutation Representation (GAP) :
s0 := (  1, 82)(  2, 81)(  3, 84)(  4, 83)(  5, 86)(  6, 85)(  7, 88)(  8, 87)
(  9, 90)( 10, 89)( 11, 92)( 12, 91)( 13, 94)( 14, 93)( 15, 96)( 16, 95)
( 17, 98)( 18, 97)( 19,100)( 20, 99)( 21,102)( 22,101)( 23,104)( 24,103)
( 25,106)( 26,105)( 27,108)( 28,107)( 29,110)( 30,109)( 31,112)( 32,111)
( 33,114)( 34,113)( 35,116)( 36,115)( 37,118)( 38,117)( 39,120)( 40,119)
( 41,122)( 42,121)( 43,124)( 44,123)( 45,126)( 46,125)( 47,128)( 48,127)
( 49,130)( 50,129)( 51,132)( 52,131)( 53,134)( 54,133)( 55,136)( 56,135)
( 57,138)( 58,137)( 59,140)( 60,139)( 61,142)( 62,141)( 63,144)( 64,143)
( 65,146)( 66,145)( 67,148)( 68,147)( 69,150)( 70,149)( 71,152)( 72,151)
( 73,154)( 74,153)( 75,156)( 76,155)( 77,158)( 78,157)( 79,160)( 80,159)
(161,242)(162,241)(163,244)(164,243)(165,246)(166,245)(167,248)(168,247)
(169,250)(170,249)(171,252)(172,251)(173,254)(174,253)(175,256)(176,255)
(177,258)(178,257)(179,260)(180,259)(181,262)(182,261)(183,264)(184,263)
(185,266)(186,265)(187,268)(188,267)(189,270)(190,269)(191,272)(192,271)
(193,274)(194,273)(195,276)(196,275)(197,278)(198,277)(199,280)(200,279)
(201,282)(202,281)(203,284)(204,283)(205,286)(206,285)(207,288)(208,287)
(209,290)(210,289)(211,292)(212,291)(213,294)(214,293)(215,296)(216,295)
(217,298)(218,297)(219,300)(220,299)(221,302)(222,301)(223,304)(224,303)
(225,306)(226,305)(227,308)(228,307)(229,310)(230,309)(231,312)(232,311)
(233,314)(234,313)(235,316)(236,315)(237,318)(238,317)(239,320)(240,319);;
s1 := (  1,161)(  2,171)(  3,174)(  4,168)(  5,165)(  6,175)(  7,170)(  8,164)
(  9,173)( 10,167)( 11,162)( 12,172)( 13,169)( 14,163)( 15,166)( 16,176)
( 17,225)( 18,235)( 19,238)( 20,232)( 21,229)( 22,239)( 23,234)( 24,228)
( 25,237)( 26,231)( 27,226)( 28,236)( 29,233)( 30,227)( 31,230)( 32,240)
( 33,209)( 34,219)( 35,222)( 36,216)( 37,213)( 38,223)( 39,218)( 40,212)
( 41,221)( 42,215)( 43,210)( 44,220)( 45,217)( 46,211)( 47,214)( 48,224)
( 49,193)( 50,203)( 51,206)( 52,200)( 53,197)( 54,207)( 55,202)( 56,196)
( 57,205)( 58,199)( 59,194)( 60,204)( 61,201)( 62,195)( 63,198)( 64,208)
( 65,177)( 66,187)( 67,190)( 68,184)( 69,181)( 70,191)( 71,186)( 72,180)
( 73,189)( 74,183)( 75,178)( 76,188)( 77,185)( 78,179)( 79,182)( 80,192)
( 81,241)( 82,251)( 83,254)( 84,248)( 85,245)( 86,255)( 87,250)( 88,244)
( 89,253)( 90,247)( 91,242)( 92,252)( 93,249)( 94,243)( 95,246)( 96,256)
( 97,305)( 98,315)( 99,318)(100,312)(101,309)(102,319)(103,314)(104,308)
(105,317)(106,311)(107,306)(108,316)(109,313)(110,307)(111,310)(112,320)
(113,289)(114,299)(115,302)(116,296)(117,293)(118,303)(119,298)(120,292)
(121,301)(122,295)(123,290)(124,300)(125,297)(126,291)(127,294)(128,304)
(129,273)(130,283)(131,286)(132,280)(133,277)(134,287)(135,282)(136,276)
(137,285)(138,279)(139,274)(140,284)(141,281)(142,275)(143,278)(144,288)
(145,257)(146,267)(147,270)(148,264)(149,261)(150,271)(151,266)(152,260)
(153,269)(154,263)(155,258)(156,268)(157,265)(158,259)(159,262)(160,272);;
s2 := (  1,193)(  2,194)(  3,196)(  4,195)(  5,198)(  6,197)(  7,199)(  8,200)
(  9,208)( 10,207)( 11,205)( 12,206)( 13,203)( 14,204)( 15,202)( 16,201)
( 17,177)( 18,178)( 19,180)( 20,179)( 21,182)( 22,181)( 23,183)( 24,184)
( 25,192)( 26,191)( 27,189)( 28,190)( 29,187)( 30,188)( 31,186)( 32,185)
( 33,161)( 34,162)( 35,164)( 36,163)( 37,166)( 38,165)( 39,167)( 40,168)
( 41,176)( 42,175)( 43,173)( 44,174)( 45,171)( 46,172)( 47,170)( 48,169)
( 49,225)( 50,226)( 51,228)( 52,227)( 53,230)( 54,229)( 55,231)( 56,232)
( 57,240)( 58,239)( 59,237)( 60,238)( 61,235)( 62,236)( 63,234)( 64,233)
( 65,209)( 66,210)( 67,212)( 68,211)( 69,214)( 70,213)( 71,215)( 72,216)
( 73,224)( 74,223)( 75,221)( 76,222)( 77,219)( 78,220)( 79,218)( 80,217)
( 81,273)( 82,274)( 83,276)( 84,275)( 85,278)( 86,277)( 87,279)( 88,280)
( 89,288)( 90,287)( 91,285)( 92,286)( 93,283)( 94,284)( 95,282)( 96,281)
( 97,257)( 98,258)( 99,260)(100,259)(101,262)(102,261)(103,263)(104,264)
(105,272)(106,271)(107,269)(108,270)(109,267)(110,268)(111,266)(112,265)
(113,241)(114,242)(115,244)(116,243)(117,246)(118,245)(119,247)(120,248)
(121,256)(122,255)(123,253)(124,254)(125,251)(126,252)(127,250)(128,249)
(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)(136,312)
(137,320)(138,319)(139,317)(140,318)(141,315)(142,316)(143,314)(144,313)
(145,289)(146,290)(147,292)(148,291)(149,294)(150,293)(151,295)(152,296)
(153,304)(154,303)(155,301)(156,302)(157,299)(158,300)(159,298)(160,297);;
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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(320)!(  1, 82)(  2, 81)(  3, 84)(  4, 83)(  5, 86)(  6, 85)(  7, 88)
(  8, 87)(  9, 90)( 10, 89)( 11, 92)( 12, 91)( 13, 94)( 14, 93)( 15, 96)
( 16, 95)( 17, 98)( 18, 97)( 19,100)( 20, 99)( 21,102)( 22,101)( 23,104)
( 24,103)( 25,106)( 26,105)( 27,108)( 28,107)( 29,110)( 30,109)( 31,112)
( 32,111)( 33,114)( 34,113)( 35,116)( 36,115)( 37,118)( 38,117)( 39,120)
( 40,119)( 41,122)( 42,121)( 43,124)( 44,123)( 45,126)( 46,125)( 47,128)
( 48,127)( 49,130)( 50,129)( 51,132)( 52,131)( 53,134)( 54,133)( 55,136)
( 56,135)( 57,138)( 58,137)( 59,140)( 60,139)( 61,142)( 62,141)( 63,144)
( 64,143)( 65,146)( 66,145)( 67,148)( 68,147)( 69,150)( 70,149)( 71,152)
( 72,151)( 73,154)( 74,153)( 75,156)( 76,155)( 77,158)( 78,157)( 79,160)
( 80,159)(161,242)(162,241)(163,244)(164,243)(165,246)(166,245)(167,248)
(168,247)(169,250)(170,249)(171,252)(172,251)(173,254)(174,253)(175,256)
(176,255)(177,258)(178,257)(179,260)(180,259)(181,262)(182,261)(183,264)
(184,263)(185,266)(186,265)(187,268)(188,267)(189,270)(190,269)(191,272)
(192,271)(193,274)(194,273)(195,276)(196,275)(197,278)(198,277)(199,280)
(200,279)(201,282)(202,281)(203,284)(204,283)(205,286)(206,285)(207,288)
(208,287)(209,290)(210,289)(211,292)(212,291)(213,294)(214,293)(215,296)
(216,295)(217,298)(218,297)(219,300)(220,299)(221,302)(222,301)(223,304)
(224,303)(225,306)(226,305)(227,308)(228,307)(229,310)(230,309)(231,312)
(232,311)(233,314)(234,313)(235,316)(236,315)(237,318)(238,317)(239,320)
(240,319);
s1 := Sym(320)!(  1,161)(  2,171)(  3,174)(  4,168)(  5,165)(  6,175)(  7,170)
(  8,164)(  9,173)( 10,167)( 11,162)( 12,172)( 13,169)( 14,163)( 15,166)
( 16,176)( 17,225)( 18,235)( 19,238)( 20,232)( 21,229)( 22,239)( 23,234)
( 24,228)( 25,237)( 26,231)( 27,226)( 28,236)( 29,233)( 30,227)( 31,230)
( 32,240)( 33,209)( 34,219)( 35,222)( 36,216)( 37,213)( 38,223)( 39,218)
( 40,212)( 41,221)( 42,215)( 43,210)( 44,220)( 45,217)( 46,211)( 47,214)
( 48,224)( 49,193)( 50,203)( 51,206)( 52,200)( 53,197)( 54,207)( 55,202)
( 56,196)( 57,205)( 58,199)( 59,194)( 60,204)( 61,201)( 62,195)( 63,198)
( 64,208)( 65,177)( 66,187)( 67,190)( 68,184)( 69,181)( 70,191)( 71,186)
( 72,180)( 73,189)( 74,183)( 75,178)( 76,188)( 77,185)( 78,179)( 79,182)
( 80,192)( 81,241)( 82,251)( 83,254)( 84,248)( 85,245)( 86,255)( 87,250)
( 88,244)( 89,253)( 90,247)( 91,242)( 92,252)( 93,249)( 94,243)( 95,246)
( 96,256)( 97,305)( 98,315)( 99,318)(100,312)(101,309)(102,319)(103,314)
(104,308)(105,317)(106,311)(107,306)(108,316)(109,313)(110,307)(111,310)
(112,320)(113,289)(114,299)(115,302)(116,296)(117,293)(118,303)(119,298)
(120,292)(121,301)(122,295)(123,290)(124,300)(125,297)(126,291)(127,294)
(128,304)(129,273)(130,283)(131,286)(132,280)(133,277)(134,287)(135,282)
(136,276)(137,285)(138,279)(139,274)(140,284)(141,281)(142,275)(143,278)
(144,288)(145,257)(146,267)(147,270)(148,264)(149,261)(150,271)(151,266)
(152,260)(153,269)(154,263)(155,258)(156,268)(157,265)(158,259)(159,262)
(160,272);
s2 := Sym(320)!(  1,193)(  2,194)(  3,196)(  4,195)(  5,198)(  6,197)(  7,199)
(  8,200)(  9,208)( 10,207)( 11,205)( 12,206)( 13,203)( 14,204)( 15,202)
( 16,201)( 17,177)( 18,178)( 19,180)( 20,179)( 21,182)( 22,181)( 23,183)
( 24,184)( 25,192)( 26,191)( 27,189)( 28,190)( 29,187)( 30,188)( 31,186)
( 32,185)( 33,161)( 34,162)( 35,164)( 36,163)( 37,166)( 38,165)( 39,167)
( 40,168)( 41,176)( 42,175)( 43,173)( 44,174)( 45,171)( 46,172)( 47,170)
( 48,169)( 49,225)( 50,226)( 51,228)( 52,227)( 53,230)( 54,229)( 55,231)
( 56,232)( 57,240)( 58,239)( 59,237)( 60,238)( 61,235)( 62,236)( 63,234)
( 64,233)( 65,209)( 66,210)( 67,212)( 68,211)( 69,214)( 70,213)( 71,215)
( 72,216)( 73,224)( 74,223)( 75,221)( 76,222)( 77,219)( 78,220)( 79,218)
( 80,217)( 81,273)( 82,274)( 83,276)( 84,275)( 85,278)( 86,277)( 87,279)
( 88,280)( 89,288)( 90,287)( 91,285)( 92,286)( 93,283)( 94,284)( 95,282)
( 96,281)( 97,257)( 98,258)( 99,260)(100,259)(101,262)(102,261)(103,263)
(104,264)(105,272)(106,271)(107,269)(108,270)(109,267)(110,268)(111,266)
(112,265)(113,241)(114,242)(115,244)(116,243)(117,246)(118,245)(119,247)
(120,248)(121,256)(122,255)(123,253)(124,254)(125,251)(126,252)(127,250)
(128,249)(129,305)(130,306)(131,308)(132,307)(133,310)(134,309)(135,311)
(136,312)(137,320)(138,319)(139,317)(140,318)(141,315)(142,316)(143,314)
(144,313)(145,289)(146,290)(147,292)(148,291)(149,294)(150,293)(151,295)
(152,296)(153,304)(154,303)(155,301)(156,302)(157,299)(158,300)(159,298)
(160,297);
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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >; 
 
References : None.
to this polytope