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Polytope of Type {2,4,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,72}*1152c
if this polytope has a name.
Group : SmallGroup(1152,154349)
Rank : 4
Schlafli Type : {2,4,72}
Number of vertices, edges, etc : 2, 4, 144, 72
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,36}*576b
   3-fold quotients : {2,4,24}*384c
   4-fold quotients : {2,4,18}*288b
   6-fold quotients : {2,4,12}*192b
   8-fold quotients : {2,4,9}*144
   12-fold quotients : {2,4,6}*96c
   24-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)(224,226)
(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)(240,242)
(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)(256,258)
(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)(272,274)
(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)(288,290);;
s2 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)
( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)
( 26, 38)( 40, 41)( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 51, 67)( 52, 69)
( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)( 58, 66)( 59, 71)( 60, 73)
( 61, 72)( 62, 74)( 75,111)( 76,113)( 77,112)( 78,114)( 79,119)( 80,121)
( 81,120)( 82,122)( 83,115)( 84,117)( 85,116)( 86,118)( 87,139)( 88,141)
( 89,140)( 90,142)( 91,135)( 92,137)( 93,136)( 94,138)( 95,143)( 96,145)
( 97,144)( 98,146)( 99,127)(100,129)(101,128)(102,130)(103,123)(104,125)
(105,124)(106,126)(107,131)(108,133)(109,132)(110,134)(147,219)(148,221)
(149,220)(150,222)(151,227)(152,229)(153,228)(154,230)(155,223)(156,225)
(157,224)(158,226)(159,247)(160,249)(161,248)(162,250)(163,243)(164,245)
(165,244)(166,246)(167,251)(168,253)(169,252)(170,254)(171,235)(172,237)
(173,236)(174,238)(175,231)(176,233)(177,232)(178,234)(179,239)(180,241)
(181,240)(182,242)(183,255)(184,257)(185,256)(186,258)(187,263)(188,265)
(189,264)(190,266)(191,259)(192,261)(193,260)(194,262)(195,283)(196,285)
(197,284)(198,286)(199,279)(200,281)(201,280)(202,282)(203,287)(204,289)
(205,288)(206,290)(207,271)(208,273)(209,272)(210,274)(211,267)(212,269)
(213,268)(214,270)(215,275)(216,277)(217,276)(218,278);;
s3 := (  3,171)(  4,174)(  5,173)(  6,172)(  7,179)(  8,182)(  9,181)( 10,180)
( 11,175)( 12,178)( 13,177)( 14,176)( 15,159)( 16,162)( 17,161)( 18,160)
( 19,167)( 20,170)( 21,169)( 22,168)( 23,163)( 24,166)( 25,165)( 26,164)
( 27,147)( 28,150)( 29,149)( 30,148)( 31,155)( 32,158)( 33,157)( 34,156)
( 35,151)( 36,154)( 37,153)( 38,152)( 39,207)( 40,210)( 41,209)( 42,208)
( 43,215)( 44,218)( 45,217)( 46,216)( 47,211)( 48,214)( 49,213)( 50,212)
( 51,195)( 52,198)( 53,197)( 54,196)( 55,203)( 56,206)( 57,205)( 58,204)
( 59,199)( 60,202)( 61,201)( 62,200)( 63,183)( 64,186)( 65,185)( 66,184)
( 67,191)( 68,194)( 69,193)( 70,192)( 71,187)( 72,190)( 73,189)( 74,188)
( 75,279)( 76,282)( 77,281)( 78,280)( 79,287)( 80,290)( 81,289)( 82,288)
( 83,283)( 84,286)( 85,285)( 86,284)( 87,267)( 88,270)( 89,269)( 90,268)
( 91,275)( 92,278)( 93,277)( 94,276)( 95,271)( 96,274)( 97,273)( 98,272)
( 99,255)(100,258)(101,257)(102,256)(103,263)(104,266)(105,265)(106,264)
(107,259)(108,262)(109,261)(110,260)(111,243)(112,246)(113,245)(114,244)
(115,251)(116,254)(117,253)(118,252)(119,247)(120,250)(121,249)(122,248)
(123,231)(124,234)(125,233)(126,232)(127,239)(128,242)(129,241)(130,240)
(131,235)(132,238)(133,237)(134,236)(135,219)(136,222)(137,221)(138,220)
(139,227)(140,230)(141,229)(142,228)(143,223)(144,226)(145,225)(146,224);;
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, s1*s2*s3*s2*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(290)!(1,2);
s1 := Sym(290)!(  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)
(224,226)(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)
(240,242)(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)
(256,258)(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)
(272,274)(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)
(288,290);
s2 := Sym(290)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)
( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)
( 25, 36)( 26, 38)( 40, 41)( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 51, 67)
( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)( 58, 66)( 59, 71)
( 60, 73)( 61, 72)( 62, 74)( 75,111)( 76,113)( 77,112)( 78,114)( 79,119)
( 80,121)( 81,120)( 82,122)( 83,115)( 84,117)( 85,116)( 86,118)( 87,139)
( 88,141)( 89,140)( 90,142)( 91,135)( 92,137)( 93,136)( 94,138)( 95,143)
( 96,145)( 97,144)( 98,146)( 99,127)(100,129)(101,128)(102,130)(103,123)
(104,125)(105,124)(106,126)(107,131)(108,133)(109,132)(110,134)(147,219)
(148,221)(149,220)(150,222)(151,227)(152,229)(153,228)(154,230)(155,223)
(156,225)(157,224)(158,226)(159,247)(160,249)(161,248)(162,250)(163,243)
(164,245)(165,244)(166,246)(167,251)(168,253)(169,252)(170,254)(171,235)
(172,237)(173,236)(174,238)(175,231)(176,233)(177,232)(178,234)(179,239)
(180,241)(181,240)(182,242)(183,255)(184,257)(185,256)(186,258)(187,263)
(188,265)(189,264)(190,266)(191,259)(192,261)(193,260)(194,262)(195,283)
(196,285)(197,284)(198,286)(199,279)(200,281)(201,280)(202,282)(203,287)
(204,289)(205,288)(206,290)(207,271)(208,273)(209,272)(210,274)(211,267)
(212,269)(213,268)(214,270)(215,275)(216,277)(217,276)(218,278);
s3 := Sym(290)!(  3,171)(  4,174)(  5,173)(  6,172)(  7,179)(  8,182)(  9,181)
( 10,180)( 11,175)( 12,178)( 13,177)( 14,176)( 15,159)( 16,162)( 17,161)
( 18,160)( 19,167)( 20,170)( 21,169)( 22,168)( 23,163)( 24,166)( 25,165)
( 26,164)( 27,147)( 28,150)( 29,149)( 30,148)( 31,155)( 32,158)( 33,157)
( 34,156)( 35,151)( 36,154)( 37,153)( 38,152)( 39,207)( 40,210)( 41,209)
( 42,208)( 43,215)( 44,218)( 45,217)( 46,216)( 47,211)( 48,214)( 49,213)
( 50,212)( 51,195)( 52,198)( 53,197)( 54,196)( 55,203)( 56,206)( 57,205)
( 58,204)( 59,199)( 60,202)( 61,201)( 62,200)( 63,183)( 64,186)( 65,185)
( 66,184)( 67,191)( 68,194)( 69,193)( 70,192)( 71,187)( 72,190)( 73,189)
( 74,188)( 75,279)( 76,282)( 77,281)( 78,280)( 79,287)( 80,290)( 81,289)
( 82,288)( 83,283)( 84,286)( 85,285)( 86,284)( 87,267)( 88,270)( 89,269)
( 90,268)( 91,275)( 92,278)( 93,277)( 94,276)( 95,271)( 96,274)( 97,273)
( 98,272)( 99,255)(100,258)(101,257)(102,256)(103,263)(104,266)(105,265)
(106,264)(107,259)(108,262)(109,261)(110,260)(111,243)(112,246)(113,245)
(114,244)(115,251)(116,254)(117,253)(118,252)(119,247)(120,250)(121,249)
(122,248)(123,231)(124,234)(125,233)(126,232)(127,239)(128,242)(129,241)
(130,240)(131,235)(132,238)(133,237)(134,236)(135,219)(136,222)(137,221)
(138,220)(139,227)(140,230)(141,229)(142,228)(143,223)(144,226)(145,225)
(146,224);
poly := sub<Sym(290)|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, 
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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