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Polytope of Type {2,4,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,5}*640
if this polytope has a name.
Group : SmallGroup(640,21537)
Rank : 4
Schlafli Type : {2,4,5}
Number of vertices, edges, etc : 2, 32, 80, 40
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,5,2} of size 1280
Vertex Figure Of :
   {2,2,4,5} of size 1280
   {3,2,4,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,5}*320
   16-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,5}*1280, {2,8,5}*1280a, {2,4,5}*1280, {2,4,10}*1280a, {2,8,5}*1280b, {2,4,10}*1280b
   3-fold covers : {6,4,5}*1920, {2,4,15}*1920
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  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)( 81,162)( 82,161)
(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)(241,322)(242,321);;
s2 := (  3,163)(  4,173)(  5,176)(  6,170)(  7,167)(  8,177)(  9,172)( 10,166)
( 11,175)( 12,169)( 13,164)( 14,174)( 15,171)( 16,165)( 17,168)( 18,178)
( 19,227)( 20,237)( 21,240)( 22,234)( 23,231)( 24,241)( 25,236)( 26,230)
( 27,239)( 28,233)( 29,228)( 30,238)( 31,235)( 32,229)( 33,232)( 34,242)
( 35,211)( 36,221)( 37,224)( 38,218)( 39,215)( 40,225)( 41,220)( 42,214)
( 43,223)( 44,217)( 45,212)( 46,222)( 47,219)( 48,213)( 49,216)( 50,226)
( 51,195)( 52,205)( 53,208)( 54,202)( 55,199)( 56,209)( 57,204)( 58,198)
( 59,207)( 60,201)( 61,196)( 62,206)( 63,203)( 64,197)( 65,200)( 66,210)
( 67,179)( 68,189)( 69,192)( 70,186)( 71,183)( 72,193)( 73,188)( 74,182)
( 75,191)( 76,185)( 77,180)( 78,190)( 79,187)( 80,181)( 81,184)( 82,194)
( 83,243)( 84,253)( 85,256)( 86,250)( 87,247)( 88,257)( 89,252)( 90,246)
( 91,255)( 92,249)( 93,244)( 94,254)( 95,251)( 96,245)( 97,248)( 98,258)
( 99,307)(100,317)(101,320)(102,314)(103,311)(104,321)(105,316)(106,310)
(107,319)(108,313)(109,308)(110,318)(111,315)(112,309)(113,312)(114,322)
(115,291)(116,301)(117,304)(118,298)(119,295)(120,305)(121,300)(122,294)
(123,303)(124,297)(125,292)(126,302)(127,299)(128,293)(129,296)(130,306)
(131,275)(132,285)(133,288)(134,282)(135,279)(136,289)(137,284)(138,278)
(139,287)(140,281)(141,276)(142,286)(143,283)(144,277)(145,280)(146,290)
(147,259)(148,269)(149,272)(150,266)(151,263)(152,273)(153,268)(154,262)
(155,271)(156,265)(157,260)(158,270)(159,267)(160,261)(161,264)(162,274);;
s3 := (  3,195)(  4,196)(  5,198)(  6,197)(  7,200)(  8,199)(  9,201)( 10,202)
( 11,210)( 12,209)( 13,207)( 14,208)( 15,205)( 16,206)( 17,204)( 18,203)
( 19,179)( 20,180)( 21,182)( 22,181)( 23,184)( 24,183)( 25,185)( 26,186)
( 27,194)( 28,193)( 29,191)( 30,192)( 31,189)( 32,190)( 33,188)( 34,187)
( 35,163)( 36,164)( 37,166)( 38,165)( 39,168)( 40,167)( 41,169)( 42,170)
( 43,178)( 44,177)( 45,175)( 46,176)( 47,173)( 48,174)( 49,172)( 50,171)
( 51,227)( 52,228)( 53,230)( 54,229)( 55,232)( 56,231)( 57,233)( 58,234)
( 59,242)( 60,241)( 61,239)( 62,240)( 63,237)( 64,238)( 65,236)( 66,235)
( 67,211)( 68,212)( 69,214)( 70,213)( 71,216)( 72,215)( 73,217)( 74,218)
( 75,226)( 76,225)( 77,223)( 78,224)( 79,221)( 80,222)( 81,220)( 82,219)
( 83,275)( 84,276)( 85,278)( 86,277)( 87,280)( 88,279)( 89,281)( 90,282)
( 91,290)( 92,289)( 93,287)( 94,288)( 95,285)( 96,286)( 97,284)( 98,283)
( 99,259)(100,260)(101,262)(102,261)(103,264)(104,263)(105,265)(106,266)
(107,274)(108,273)(109,271)(110,272)(111,269)(112,270)(113,268)(114,267)
(115,243)(116,244)(117,246)(118,245)(119,248)(120,247)(121,249)(122,250)
(123,258)(124,257)(125,255)(126,256)(127,253)(128,254)(129,252)(130,251)
(131,307)(132,308)(133,310)(134,309)(135,312)(136,311)(137,313)(138,314)
(139,322)(140,321)(141,319)(142,320)(143,317)(144,318)(145,316)(146,315)
(147,291)(148,292)(149,294)(150,293)(151,296)(152,295)(153,297)(154,298)
(155,306)(156,305)(157,303)(158,304)(159,301)(160,302)(161,300)(162,299);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(322)!(1,2);
s1 := Sym(322)!(  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)( 81,162)
( 82,161)(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)(241,322)
(242,321);
s2 := Sym(322)!(  3,163)(  4,173)(  5,176)(  6,170)(  7,167)(  8,177)(  9,172)
( 10,166)( 11,175)( 12,169)( 13,164)( 14,174)( 15,171)( 16,165)( 17,168)
( 18,178)( 19,227)( 20,237)( 21,240)( 22,234)( 23,231)( 24,241)( 25,236)
( 26,230)( 27,239)( 28,233)( 29,228)( 30,238)( 31,235)( 32,229)( 33,232)
( 34,242)( 35,211)( 36,221)( 37,224)( 38,218)( 39,215)( 40,225)( 41,220)
( 42,214)( 43,223)( 44,217)( 45,212)( 46,222)( 47,219)( 48,213)( 49,216)
( 50,226)( 51,195)( 52,205)( 53,208)( 54,202)( 55,199)( 56,209)( 57,204)
( 58,198)( 59,207)( 60,201)( 61,196)( 62,206)( 63,203)( 64,197)( 65,200)
( 66,210)( 67,179)( 68,189)( 69,192)( 70,186)( 71,183)( 72,193)( 73,188)
( 74,182)( 75,191)( 76,185)( 77,180)( 78,190)( 79,187)( 80,181)( 81,184)
( 82,194)( 83,243)( 84,253)( 85,256)( 86,250)( 87,247)( 88,257)( 89,252)
( 90,246)( 91,255)( 92,249)( 93,244)( 94,254)( 95,251)( 96,245)( 97,248)
( 98,258)( 99,307)(100,317)(101,320)(102,314)(103,311)(104,321)(105,316)
(106,310)(107,319)(108,313)(109,308)(110,318)(111,315)(112,309)(113,312)
(114,322)(115,291)(116,301)(117,304)(118,298)(119,295)(120,305)(121,300)
(122,294)(123,303)(124,297)(125,292)(126,302)(127,299)(128,293)(129,296)
(130,306)(131,275)(132,285)(133,288)(134,282)(135,279)(136,289)(137,284)
(138,278)(139,287)(140,281)(141,276)(142,286)(143,283)(144,277)(145,280)
(146,290)(147,259)(148,269)(149,272)(150,266)(151,263)(152,273)(153,268)
(154,262)(155,271)(156,265)(157,260)(158,270)(159,267)(160,261)(161,264)
(162,274);
s3 := Sym(322)!(  3,195)(  4,196)(  5,198)(  6,197)(  7,200)(  8,199)(  9,201)
( 10,202)( 11,210)( 12,209)( 13,207)( 14,208)( 15,205)( 16,206)( 17,204)
( 18,203)( 19,179)( 20,180)( 21,182)( 22,181)( 23,184)( 24,183)( 25,185)
( 26,186)( 27,194)( 28,193)( 29,191)( 30,192)( 31,189)( 32,190)( 33,188)
( 34,187)( 35,163)( 36,164)( 37,166)( 38,165)( 39,168)( 40,167)( 41,169)
( 42,170)( 43,178)( 44,177)( 45,175)( 46,176)( 47,173)( 48,174)( 49,172)
( 50,171)( 51,227)( 52,228)( 53,230)( 54,229)( 55,232)( 56,231)( 57,233)
( 58,234)( 59,242)( 60,241)( 61,239)( 62,240)( 63,237)( 64,238)( 65,236)
( 66,235)( 67,211)( 68,212)( 69,214)( 70,213)( 71,216)( 72,215)( 73,217)
( 74,218)( 75,226)( 76,225)( 77,223)( 78,224)( 79,221)( 80,222)( 81,220)
( 82,219)( 83,275)( 84,276)( 85,278)( 86,277)( 87,280)( 88,279)( 89,281)
( 90,282)( 91,290)( 92,289)( 93,287)( 94,288)( 95,285)( 96,286)( 97,284)
( 98,283)( 99,259)(100,260)(101,262)(102,261)(103,264)(104,263)(105,265)
(106,266)(107,274)(108,273)(109,271)(110,272)(111,269)(112,270)(113,268)
(114,267)(115,243)(116,244)(117,246)(118,245)(119,248)(120,247)(121,249)
(122,250)(123,258)(124,257)(125,255)(126,256)(127,253)(128,254)(129,252)
(130,251)(131,307)(132,308)(133,310)(134,309)(135,312)(136,311)(137,313)
(138,314)(139,322)(140,321)(141,319)(142,320)(143,317)(144,318)(145,316)
(146,315)(147,291)(148,292)(149,294)(150,293)(151,296)(152,295)(153,297)
(154,298)(155,306)(156,305)(157,303)(158,304)(159,301)(160,302)(161,300)
(162,299);
poly := sub<Sym(322)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >; 
 

to this polytope