Polytope of Type {72,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,4}*1152f
if this polytope has a name.
Group : SmallGroup(1152,154380)
Rank : 3
Schlafli Type : {72,4}
Number of vertices, edges, etc : 144, 288, 8
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,4}*576c
   3-fold quotients : {24,4}*384f
   4-fold quotients : {18,4}*288
   6-fold quotients : {12,4}*192c
   8-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c
   12-fold quotients : {6,4}*96
   16-fold quotients : {9,4}*72, {18,2}*72
   24-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   32-fold quotients : {9,2}*36
   48-fold quotients : {3,4}*24, {6,2}*24
   96-fold quotients : {3,2}*12
   144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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