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