Polytope of Type {4,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,72}*576c
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}*192c
   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 : {4,144}*1152c, {4,144}*1152d, {4,72}*1152c
   3-fold covers : {4,216}*1728c
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)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288);;
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*s0*s1*s2*s0*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*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(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)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288);
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*s0*s1*s2*s0*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*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 >;

References : None.
to this polytope