Polytope of Type {4,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,72}*1152f
if this polytope has a name.
Group : SmallGroup(1152,154380)
Rank : 3
Schlafli Type : {4,72}
Number of vertices, edges, etc : 8, 288, 144
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 8
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}*576c
   3-fold quotients : {4,24}*384f
   4-fold quotients : {4,18}*288
   6-fold quotients : {4,12}*192c
   8-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
   12-fold quotients : {4,6}*96
   16-fold quotients : {4,9}*72, {2,18}*72
   24-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {2,9}*36
   48-fold quotients : {4,3}*24, {2,6}*24
   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 := (  1,149)(  2,150)(  3,152)(  4,151)(  5,146)(  6,145)(  7,147)(  8,148)
(  9,157)( 10,158)( 11,160)( 12,159)( 13,154)( 14,153)( 15,155)( 16,156)
( 17,165)( 18,166)( 19,168)( 20,167)( 21,162)( 22,161)( 23,163)( 24,164)
( 25,173)( 26,174)( 27,176)( 28,175)( 29,170)( 30,169)( 31,171)( 32,172)
( 33,181)( 34,182)( 35,184)( 36,183)( 37,178)( 38,177)( 39,179)( 40,180)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,186)( 46,185)( 47,187)( 48,188)
( 49,197)( 50,198)( 51,200)( 52,199)( 53,194)( 54,193)( 55,195)( 56,196)
( 57,205)( 58,206)( 59,208)( 60,207)( 61,202)( 62,201)( 63,203)( 64,204)
( 65,213)( 66,214)( 67,216)( 68,215)( 69,210)( 70,209)( 71,211)( 72,212)
( 73,222)( 74,221)( 75,223)( 76,224)( 77,217)( 78,218)( 79,220)( 80,219)
( 81,230)( 82,229)( 83,231)( 84,232)( 85,225)( 86,226)( 87,228)( 88,227)
( 89,238)( 90,237)( 91,239)( 92,240)( 93,233)( 94,234)( 95,236)( 96,235)
( 97,246)( 98,245)( 99,247)(100,248)(101,241)(102,242)(103,244)(104,243)
(105,254)(106,253)(107,255)(108,256)(109,249)(110,250)(111,252)(112,251)
(113,262)(114,261)(115,263)(116,264)(117,257)(118,258)(119,260)(120,259)
(121,270)(122,269)(123,271)(124,272)(125,265)(126,266)(127,268)(128,267)
(129,278)(130,277)(131,279)(132,280)(133,273)(134,274)(135,276)(136,275)
(137,286)(138,285)(139,287)(140,288)(141,281)(142,282)(143,284)(144,283);;
s1 := (  3,  4)(  5,  7)(  6,  8)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)
( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)
( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)( 45, 55)
( 46, 56)( 47, 53)( 48, 54)( 73, 74)( 77, 80)( 78, 79)( 81, 90)( 82, 89)
( 83, 91)( 84, 92)( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 97,138)( 98,137)
( 99,139)(100,140)(101,144)(102,143)(103,142)(104,141)(105,130)(106,129)
(107,131)(108,132)(109,136)(110,135)(111,134)(112,133)(113,122)(114,121)
(115,123)(116,124)(117,128)(118,127)(119,126)(120,125)(145,218)(146,217)
(147,219)(148,220)(149,224)(150,223)(151,222)(152,221)(153,234)(154,233)
(155,235)(156,236)(157,240)(158,239)(159,238)(160,237)(161,226)(162,225)
(163,227)(164,228)(165,232)(166,231)(167,230)(168,229)(169,282)(170,281)
(171,283)(172,284)(173,288)(174,287)(175,286)(176,285)(177,274)(178,273)
(179,275)(180,276)(181,280)(182,279)(183,278)(184,277)(185,266)(186,265)
(187,267)(188,268)(189,272)(190,271)(191,270)(192,269)(193,258)(194,257)
(195,259)(196,260)(197,264)(198,263)(199,262)(200,261)(201,250)(202,249)
(203,251)(204,252)(205,256)(206,255)(207,254)(208,253)(209,242)(210,241)
(211,243)(212,244)(213,248)(214,247)(215,246)(216,245);;
s2 := (  1,169)(  2,170)(  3,175)(  4,176)(  5,174)(  6,173)(  7,171)(  8,172)
(  9,185)( 10,186)( 11,191)( 12,192)( 13,190)( 14,189)( 15,187)( 16,188)
( 17,177)( 18,178)( 19,183)( 20,184)( 21,182)( 22,181)( 23,179)( 24,180)
( 25,145)( 26,146)( 27,151)( 28,152)( 29,150)( 30,149)( 31,147)( 32,148)
( 33,161)( 34,162)( 35,167)( 36,168)( 37,166)( 38,165)( 39,163)( 40,164)
( 41,153)( 42,154)( 43,159)( 44,160)( 45,158)( 46,157)( 47,155)( 48,156)
( 49,209)( 50,210)( 51,215)( 52,216)( 53,214)( 54,213)( 55,211)( 56,212)
( 57,201)( 58,202)( 59,207)( 60,208)( 61,206)( 62,205)( 63,203)( 64,204)
( 65,193)( 66,194)( 67,199)( 68,200)( 69,198)( 70,197)( 71,195)( 72,196)
( 73,242)( 74,241)( 75,248)( 76,247)( 77,245)( 78,246)( 79,244)( 80,243)
( 81,258)( 82,257)( 83,264)( 84,263)( 85,261)( 86,262)( 87,260)( 88,259)
( 89,250)( 90,249)( 91,256)( 92,255)( 93,253)( 94,254)( 95,252)( 96,251)
( 97,218)( 98,217)( 99,224)(100,223)(101,221)(102,222)(103,220)(104,219)
(105,234)(106,233)(107,240)(108,239)(109,237)(110,238)(111,236)(112,235)
(113,226)(114,225)(115,232)(116,231)(117,229)(118,230)(119,228)(120,227)
(121,282)(122,281)(123,288)(124,287)(125,285)(126,286)(127,284)(128,283)
(129,274)(130,273)(131,280)(132,279)(133,277)(134,278)(135,276)(136,275)
(137,266)(138,265)(139,272)(140,271)(141,269)(142,270)(143,268)(144,267);;
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, 
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*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope