Polytope of Type {4,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,72}*1152d
if this polytope has a name.
Group : SmallGroup(1152,154353)
Rank : 3
Schlafli Type : {4,72}
Number of vertices, edges, etc : 8, 288, 144
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {4,36}*576b
   3-fold quotients : {4,24}*384d
   4-fold quotients : {4,36}*288b, {4,36}*288c, {4,18}*288
   6-fold quotients : {4,12}*192b
   8-fold quotients : {2,36}*144, {4,9}*144, {4,18}*144b, {4,18}*144c
   12-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
   16-fold quotients : {4,9}*72, {2,18}*72
   24-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {2,9}*36
   48-fold quotients : {4,3}*24, {2,6}*24
   72-fold quotients : {2,4}*16
   96-fold quotients : {2,3}*12
   144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(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 := (  1,145)(  2,146)(  3,148)(  4,147)(  5,153)(  6,154)(  7,156)(  8,155)
(  9,149)( 10,150)( 11,152)( 12,151)( 13,177)( 14,178)( 15,180)( 16,179)
( 17,173)( 18,174)( 19,176)( 20,175)( 21,169)( 22,170)( 23,172)( 24,171)
( 25,165)( 26,166)( 27,168)( 28,167)( 29,161)( 30,162)( 31,164)( 32,163)
( 33,157)( 34,158)( 35,160)( 36,159)( 37,181)( 38,182)( 39,184)( 40,183)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)( 48,187)
( 49,213)( 50,214)( 51,216)( 52,215)( 53,209)( 54,210)( 55,212)( 56,211)
( 57,205)( 58,206)( 59,208)( 60,207)( 61,201)( 62,202)( 63,204)( 64,203)
( 65,197)( 66,198)( 67,200)( 68,199)( 69,193)( 70,194)( 71,196)( 72,195)
( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)( 80,263)
( 81,257)( 82,258)( 83,260)( 84,259)( 85,285)( 86,286)( 87,288)( 88,287)
( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)( 96,279)
( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)(104,271)
(105,265)(106,266)(107,268)(108,267)(109,217)(110,218)(111,220)(112,219)
(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)(120,223)
(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)(128,247)
(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)(136,239)
(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231);;
s2 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)(  8, 22)
(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)( 28, 34)
( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)( 63, 71)
( 64, 70)( 66, 68)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)( 78,132)
( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)( 86,112)
( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)
( 95,115)( 96,114)( 97,141)( 98,144)( 99,143)(100,142)(101,137)(102,140)
(103,139)(104,138)(105,133)(106,136)(107,135)(108,134)(145,229)(146,232)
(147,231)(148,230)(149,237)(150,240)(151,239)(152,238)(153,233)(154,236)
(155,235)(156,234)(157,217)(158,220)(159,219)(160,218)(161,225)(162,228)
(163,227)(164,226)(165,221)(166,224)(167,223)(168,222)(169,249)(170,252)
(171,251)(172,250)(173,245)(174,248)(175,247)(176,246)(177,241)(178,244)
(179,243)(180,242)(181,265)(182,268)(183,267)(184,266)(185,273)(186,276)
(187,275)(188,274)(189,269)(190,272)(191,271)(192,270)(193,253)(194,256)
(195,255)(196,254)(197,261)(198,264)(199,263)(200,262)(201,257)(202,260)
(203,259)(204,258)(205,285)(206,288)(207,287)(208,286)(209,281)(210,284)
(211,283)(212,282)(213,277)(214,280)(215,279)(216,278);;
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*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*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*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(288)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(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)!(  1,145)(  2,146)(  3,148)(  4,147)(  5,153)(  6,154)(  7,156)
(  8,155)(  9,149)( 10,150)( 11,152)( 12,151)( 13,177)( 14,178)( 15,180)
( 16,179)( 17,173)( 18,174)( 19,176)( 20,175)( 21,169)( 22,170)( 23,172)
( 24,171)( 25,165)( 26,166)( 27,168)( 28,167)( 29,161)( 30,162)( 31,164)
( 32,163)( 33,157)( 34,158)( 35,160)( 36,159)( 37,181)( 38,182)( 39,184)
( 40,183)( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)
( 48,187)( 49,213)( 50,214)( 51,216)( 52,215)( 53,209)( 54,210)( 55,212)
( 56,211)( 57,205)( 58,206)( 59,208)( 60,207)( 61,201)( 62,202)( 63,204)
( 64,203)( 65,197)( 66,198)( 67,200)( 68,199)( 69,193)( 70,194)( 71,196)
( 72,195)( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)
( 80,263)( 81,257)( 82,258)( 83,260)( 84,259)( 85,285)( 86,286)( 87,288)
( 88,287)( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)
( 96,279)( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)
(104,271)(105,265)(106,266)(107,268)(108,267)(109,217)(110,218)(111,220)
(112,219)(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)
(120,223)(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)
(128,247)(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)
(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)
(144,231);
s2 := Sym(288)!(  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)
(  8, 22)(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)
( 28, 34)( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)
( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)
( 63, 71)( 64, 70)( 66, 68)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)
( 78,132)( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)
( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)
( 94,116)( 95,115)( 96,114)( 97,141)( 98,144)( 99,143)(100,142)(101,137)
(102,140)(103,139)(104,138)(105,133)(106,136)(107,135)(108,134)(145,229)
(146,232)(147,231)(148,230)(149,237)(150,240)(151,239)(152,238)(153,233)
(154,236)(155,235)(156,234)(157,217)(158,220)(159,219)(160,218)(161,225)
(162,228)(163,227)(164,226)(165,221)(166,224)(167,223)(168,222)(169,249)
(170,252)(171,251)(172,250)(173,245)(174,248)(175,247)(176,246)(177,241)
(178,244)(179,243)(180,242)(181,265)(182,268)(183,267)(184,266)(185,273)
(186,276)(187,275)(188,274)(189,269)(190,272)(191,271)(192,270)(193,253)
(194,256)(195,255)(196,254)(197,261)(198,264)(199,263)(200,262)(201,257)
(202,260)(203,259)(204,258)(205,285)(206,288)(207,287)(208,286)(209,281)
(210,284)(211,283)(212,282)(213,277)(214,280)(215,279)(216,278);
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*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*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*s0*s1*s2*s1*s2*s1 >;

References : None.
to this polytope