Polytope of Type {2,72,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,72,4}*1152d
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}*384d
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, 42)( 4, 41)( 5, 40)( 6, 39)( 7, 46)( 8, 45)( 9, 44)( 10, 43)
( 11, 50)( 12, 49)( 13, 48)( 14, 47)( 15, 54)( 16, 53)( 17, 52)( 18, 51)
( 19, 58)( 20, 57)( 21, 56)( 22, 55)( 23, 62)( 24, 61)( 25, 60)( 26, 59)
( 27, 66)( 28, 65)( 29, 64)( 30, 63)( 31, 70)( 32, 69)( 33, 68)( 34, 67)
( 35, 74)( 36, 73)( 37, 72)( 38, 71)( 75,114)( 76,113)( 77,112)( 78,111)
( 79,118)( 80,117)( 81,116)( 82,115)( 83,122)( 84,121)( 85,120)( 86,119)
( 87,126)( 88,125)( 89,124)( 90,123)( 91,130)( 92,129)( 93,128)( 94,127)
( 95,134)( 96,133)( 97,132)( 98,131)( 99,138)(100,137)(101,136)(102,135)
(103,142)(104,141)(105,140)(106,139)(107,146)(108,145)(109,144)(110,143)
(147,186)(148,185)(149,184)(150,183)(151,190)(152,189)(153,188)(154,187)
(155,194)(156,193)(157,192)(158,191)(159,198)(160,197)(161,196)(162,195)
(163,202)(164,201)(165,200)(166,199)(167,206)(168,205)(169,204)(170,203)
(171,210)(172,209)(173,208)(174,207)(175,214)(176,213)(177,212)(178,211)
(179,218)(180,217)(181,216)(182,215)(219,258)(220,257)(221,256)(222,255)
(223,262)(224,261)(225,260)(226,259)(227,266)(228,265)(229,264)(230,263)
(231,270)(232,269)(233,268)(234,267)(235,274)(236,273)(237,272)(238,271)
(239,278)(240,277)(241,276)(242,275)(243,282)(244,281)(245,280)(246,279)
(247,286)(248,285)(249,284)(250,283)(251,290)(252,289)(253,288)(254,287);;
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, s1*s2*s3*s2*s1*s2*s1*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*s3*s1*s2*s3*s1*s2*s3 ];;
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, 42)( 4, 41)( 5, 40)( 6, 39)( 7, 46)( 8, 45)( 9, 44)
( 10, 43)( 11, 50)( 12, 49)( 13, 48)( 14, 47)( 15, 54)( 16, 53)( 17, 52)
( 18, 51)( 19, 58)( 20, 57)( 21, 56)( 22, 55)( 23, 62)( 24, 61)( 25, 60)
( 26, 59)( 27, 66)( 28, 65)( 29, 64)( 30, 63)( 31, 70)( 32, 69)( 33, 68)
( 34, 67)( 35, 74)( 36, 73)( 37, 72)( 38, 71)( 75,114)( 76,113)( 77,112)
( 78,111)( 79,118)( 80,117)( 81,116)( 82,115)( 83,122)( 84,121)( 85,120)
( 86,119)( 87,126)( 88,125)( 89,124)( 90,123)( 91,130)( 92,129)( 93,128)
( 94,127)( 95,134)( 96,133)( 97,132)( 98,131)( 99,138)(100,137)(101,136)
(102,135)(103,142)(104,141)(105,140)(106,139)(107,146)(108,145)(109,144)
(110,143)(147,186)(148,185)(149,184)(150,183)(151,190)(152,189)(153,188)
(154,187)(155,194)(156,193)(157,192)(158,191)(159,198)(160,197)(161,196)
(162,195)(163,202)(164,201)(165,200)(166,199)(167,206)(168,205)(169,204)
(170,203)(171,210)(172,209)(173,208)(174,207)(175,214)(176,213)(177,212)
(178,211)(179,218)(180,217)(181,216)(182,215)(219,258)(220,257)(221,256)
(222,255)(223,262)(224,261)(225,260)(226,259)(227,266)(228,265)(229,264)
(230,263)(231,270)(232,269)(233,268)(234,267)(235,274)(236,273)(237,272)
(238,271)(239,278)(240,277)(241,276)(242,275)(243,282)(244,281)(245,280)
(246,279)(247,286)(248,285)(249,284)(250,283)(251,290)(252,289)(253,288)
(254,287);
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,
s1*s2*s3*s2*s1*s2*s1*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*s3*s1*s2*s3*s1*s2*s3 >;
to this polytope