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