Polytope of Type {72,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,4}*576c
if this polytope has a name.
Group : SmallGroup(576,1386)
Rank : 3
Schlafli Type : {72,4}
Number of vertices, edges, etc : 72, 144, 4
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {72,4,2} of size 1152
Vertex Figure Of :
   {2,72,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,4}*288b
   3-fold quotients : {24,4}*192c
   4-fold quotients : {18,4}*144b
   6-fold quotients : {12,4}*96b
   8-fold quotients : {9,4}*72
   12-fold quotients : {6,4}*48c
   24-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {144,4}*1152c, {144,4}*1152d, {72,4}*1152c
   3-fold covers : {216,4}*1728c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope