Polytope of Type {4,72}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,72}*576d
if this polytope has a name.
Group : SmallGroup(576,1386)
Rank : 3
Schlafli Type : {4,72}
Number of vertices, edges, etc : 4, 144, 72
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,72,2} of size 1152
Vertex Figure Of :
{2,4,72} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,36}*288b
3-fold quotients : {4,24}*192d
4-fold quotients : {4,18}*144b
6-fold quotients : {4,12}*96b
8-fold quotients : {4,9}*72
12-fold quotients : {4,6}*48c
24-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,72}*1152c
3-fold covers : {4,216}*1728d
Permutation Representation (GAP) :
s0 := ( 1, 39)( 2, 40)( 3, 37)( 4, 38)( 5, 43)( 6, 44)( 7, 41)( 8, 42)
( 9, 47)( 10, 48)( 11, 45)( 12, 46)( 13, 51)( 14, 52)( 15, 49)( 16, 50)
( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 59)( 22, 60)( 23, 57)( 24, 58)
( 25, 63)( 26, 64)( 27, 61)( 28, 62)( 29, 67)( 30, 68)( 31, 65)( 32, 66)
( 33, 71)( 34, 72)( 35, 69)( 36, 70)( 73,111)( 74,112)( 75,109)( 76,110)
( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)( 84,118)
( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)( 92,126)
( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)(100,134)
(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)(108,142)
(145,183)(146,184)(147,181)(148,182)(149,187)(150,188)(151,185)(152,186)
(153,191)(154,192)(155,189)(156,190)(157,195)(158,196)(159,193)(160,194)
(161,199)(162,200)(163,197)(164,198)(165,203)(166,204)(167,201)(168,202)
(169,207)(170,208)(171,205)(172,206)(173,211)(174,212)(175,209)(176,210)
(177,215)(178,216)(179,213)(180,214)(217,255)(218,256)(219,253)(220,254)
(221,259)(222,260)(223,257)(224,258)(225,263)(226,264)(227,261)(228,262)
(229,267)(230,268)(231,265)(232,266)(233,271)(234,272)(235,269)(236,270)
(237,275)(238,276)(239,273)(240,274)(241,279)(242,280)(243,277)(244,278)
(245,283)(246,284)(247,281)(248,282)(249,287)(250,288)(251,285)(252,286);;
s1 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)( 78,119)
( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,137)( 86,139)
( 87,138)( 88,140)( 89,133)( 90,135)( 91,134)( 92,136)( 93,141)( 94,143)
( 95,142)( 96,144)( 97,125)( 98,127)( 99,126)(100,128)(101,121)(102,123)
(103,122)(104,124)(105,129)(106,131)(107,130)(108,132)(145,217)(146,219)
(147,218)(148,220)(149,225)(150,227)(151,226)(152,228)(153,221)(154,223)
(155,222)(156,224)(157,245)(158,247)(159,246)(160,248)(161,241)(162,243)
(163,242)(164,244)(165,249)(166,251)(167,250)(168,252)(169,233)(170,235)
(171,234)(172,236)(173,229)(174,231)(175,230)(176,232)(177,237)(178,239)
(179,238)(180,240)(181,253)(182,255)(183,254)(184,256)(185,261)(186,263)
(187,262)(188,264)(189,257)(190,259)(191,258)(192,260)(193,281)(194,283)
(195,282)(196,284)(197,277)(198,279)(199,278)(200,280)(201,285)(202,287)
(203,286)(204,288)(205,269)(206,271)(207,270)(208,272)(209,265)(210,267)
(211,266)(212,268)(213,273)(214,275)(215,274)(216,276);;
s2 := ( 1,169)( 2,172)( 3,171)( 4,170)( 5,177)( 6,180)( 7,179)( 8,178)
( 9,173)( 10,176)( 11,175)( 12,174)( 13,157)( 14,160)( 15,159)( 16,158)
( 17,165)( 18,168)( 19,167)( 20,166)( 21,161)( 22,164)( 23,163)( 24,162)
( 25,145)( 26,148)( 27,147)( 28,146)( 29,153)( 30,156)( 31,155)( 32,154)
( 33,149)( 34,152)( 35,151)( 36,150)( 37,205)( 38,208)( 39,207)( 40,206)
( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)( 48,210)
( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)( 56,202)
( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)( 64,182)
( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)( 72,186)
( 73,277)( 74,280)( 75,279)( 76,278)( 77,285)( 78,288)( 79,287)( 80,286)
( 81,281)( 82,284)( 83,283)( 84,282)( 85,265)( 86,268)( 87,267)( 88,266)
( 89,273)( 90,276)( 91,275)( 92,274)( 93,269)( 94,272)( 95,271)( 96,270)
( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)(104,262)
(105,257)(106,260)(107,259)(108,258)(109,241)(110,244)(111,243)(112,242)
(113,249)(114,252)(115,251)(116,250)(117,245)(118,248)(119,247)(120,246)
(121,229)(122,232)(123,231)(124,230)(125,237)(126,240)(127,239)(128,238)
(129,233)(130,236)(131,235)(132,234)(133,217)(134,220)(135,219)(136,218)
(137,225)(138,228)(139,227)(140,226)(141,221)(142,224)(143,223)(144,222);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
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*s0*s2*s1*s0*s2*s1*s0*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(288)!( 1, 39)( 2, 40)( 3, 37)( 4, 38)( 5, 43)( 6, 44)( 7, 41)
( 8, 42)( 9, 47)( 10, 48)( 11, 45)( 12, 46)( 13, 51)( 14, 52)( 15, 49)
( 16, 50)( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 59)( 22, 60)( 23, 57)
( 24, 58)( 25, 63)( 26, 64)( 27, 61)( 28, 62)( 29, 67)( 30, 68)( 31, 65)
( 32, 66)( 33, 71)( 34, 72)( 35, 69)( 36, 70)( 73,111)( 74,112)( 75,109)
( 76,110)( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)
( 84,118)( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)
( 92,126)( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)
(100,134)(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)
(108,142)(145,183)(146,184)(147,181)(148,182)(149,187)(150,188)(151,185)
(152,186)(153,191)(154,192)(155,189)(156,190)(157,195)(158,196)(159,193)
(160,194)(161,199)(162,200)(163,197)(164,198)(165,203)(166,204)(167,201)
(168,202)(169,207)(170,208)(171,205)(172,206)(173,211)(174,212)(175,209)
(176,210)(177,215)(178,216)(179,213)(180,214)(217,255)(218,256)(219,253)
(220,254)(221,259)(222,260)(223,257)(224,258)(225,263)(226,264)(227,261)
(228,262)(229,267)(230,268)(231,265)(232,266)(233,271)(234,272)(235,269)
(236,270)(237,275)(238,276)(239,273)(240,274)(241,279)(242,280)(243,277)
(244,278)(245,283)(246,284)(247,281)(248,282)(249,287)(250,288)(251,285)
(252,286);
s1 := Sym(288)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)
( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)
( 58, 71)( 59, 70)( 60, 72)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)
( 78,119)( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,137)
( 86,139)( 87,138)( 88,140)( 89,133)( 90,135)( 91,134)( 92,136)( 93,141)
( 94,143)( 95,142)( 96,144)( 97,125)( 98,127)( 99,126)(100,128)(101,121)
(102,123)(103,122)(104,124)(105,129)(106,131)(107,130)(108,132)(145,217)
(146,219)(147,218)(148,220)(149,225)(150,227)(151,226)(152,228)(153,221)
(154,223)(155,222)(156,224)(157,245)(158,247)(159,246)(160,248)(161,241)
(162,243)(163,242)(164,244)(165,249)(166,251)(167,250)(168,252)(169,233)
(170,235)(171,234)(172,236)(173,229)(174,231)(175,230)(176,232)(177,237)
(178,239)(179,238)(180,240)(181,253)(182,255)(183,254)(184,256)(185,261)
(186,263)(187,262)(188,264)(189,257)(190,259)(191,258)(192,260)(193,281)
(194,283)(195,282)(196,284)(197,277)(198,279)(199,278)(200,280)(201,285)
(202,287)(203,286)(204,288)(205,269)(206,271)(207,270)(208,272)(209,265)
(210,267)(211,266)(212,268)(213,273)(214,275)(215,274)(216,276);
s2 := Sym(288)!( 1,169)( 2,172)( 3,171)( 4,170)( 5,177)( 6,180)( 7,179)
( 8,178)( 9,173)( 10,176)( 11,175)( 12,174)( 13,157)( 14,160)( 15,159)
( 16,158)( 17,165)( 18,168)( 19,167)( 20,166)( 21,161)( 22,164)( 23,163)
( 24,162)( 25,145)( 26,148)( 27,147)( 28,146)( 29,153)( 30,156)( 31,155)
( 32,154)( 33,149)( 34,152)( 35,151)( 36,150)( 37,205)( 38,208)( 39,207)
( 40,206)( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)
( 48,210)( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)
( 56,202)( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)
( 64,182)( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)
( 72,186)( 73,277)( 74,280)( 75,279)( 76,278)( 77,285)( 78,288)( 79,287)
( 80,286)( 81,281)( 82,284)( 83,283)( 84,282)( 85,265)( 86,268)( 87,267)
( 88,266)( 89,273)( 90,276)( 91,275)( 92,274)( 93,269)( 94,272)( 95,271)
( 96,270)( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)
(104,262)(105,257)(106,260)(107,259)(108,258)(109,241)(110,244)(111,243)
(112,242)(113,249)(114,252)(115,251)(116,250)(117,245)(118,248)(119,247)
(120,246)(121,229)(122,232)(123,231)(124,230)(125,237)(126,240)(127,239)
(128,238)(129,233)(130,236)(131,235)(132,234)(133,217)(134,220)(135,219)
(136,218)(137,225)(138,228)(139,227)(140,226)(141,221)(142,224)(143,223)
(144,222);
poly := sub<Sym(288)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
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*s0*s2*s1*s0*s2*s1*s0*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 >;
References : None.
to this polytope