Polytope of Type {4,66}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,66}*528b
if this polytope has a name.
Group : SmallGroup(528,162)
Rank : 3
Schlafli Type : {4,66}
Number of vertices, edges, etc : 4, 132, 66
Order of s₀s₁s₂ : 66
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,66,2} of size 1056
Vertex Figure Of :
   {2,4,66} of size 1056
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,33}*264
   11-fold quotients : {4,6}*48c
   22-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,132}*1056b, {4,132}*1056c, {4,66}*1056
   3-fold covers : {4,198}*1584b
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);;
s1 := (  2,  3)(  5, 41)(  6, 43)(  7, 42)(  8, 44)(  9, 37)( 10, 39)( 11, 38)
( 12, 40)( 13, 33)( 14, 35)( 15, 34)( 16, 36)( 17, 29)( 18, 31)( 19, 30)
( 20, 32)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 45, 89)( 46, 91)( 47, 90)
( 48, 92)( 49,129)( 50,131)( 51,130)( 52,132)( 53,125)( 54,127)( 55,126)
( 56,128)( 57,121)( 58,123)( 59,122)( 60,124)( 61,117)( 62,119)( 63,118)
( 64,120)( 65,113)( 66,115)( 67,114)( 68,116)( 69,109)( 70,111)( 71,110)
( 72,112)( 73,105)( 74,107)( 75,106)( 76,108)( 77,101)( 78,103)( 79,102)
( 80,104)( 81, 97)( 82, 99)( 83, 98)( 84,100)( 85, 93)( 86, 95)( 87, 94)
( 88, 96)(134,135)(137,173)(138,175)(139,174)(140,176)(141,169)(142,171)
(143,170)(144,172)(145,165)(146,167)(147,166)(148,168)(149,161)(150,163)
(151,162)(152,164)(153,157)(154,159)(155,158)(156,160)(177,221)(178,223)
(179,222)(180,224)(181,261)(182,263)(183,262)(184,264)(185,257)(186,259)
(187,258)(188,260)(189,253)(190,255)(191,254)(192,256)(193,249)(194,251)
(195,250)(196,252)(197,245)(198,247)(199,246)(200,248)(201,241)(202,243)
(203,242)(204,244)(205,237)(206,239)(207,238)(208,240)(209,233)(210,235)
(211,234)(212,236)(213,229)(214,231)(215,230)(216,232)(217,225)(218,227)
(219,226)(220,228);;
s2 := (  1,225)(  2,228)(  3,227)(  4,226)(  5,221)(  6,224)(  7,223)(  8,222)
(  9,261)( 10,264)( 11,263)( 12,262)( 13,257)( 14,260)( 15,259)( 16,258)
( 17,253)( 18,256)( 19,255)( 20,254)( 21,249)( 22,252)( 23,251)( 24,250)
( 25,245)( 26,248)( 27,247)( 28,246)( 29,241)( 30,244)( 31,243)( 32,242)
( 33,237)( 34,240)( 35,239)( 36,238)( 37,233)( 38,236)( 39,235)( 40,234)
( 41,229)( 42,232)( 43,231)( 44,230)( 45,181)( 46,184)( 47,183)( 48,182)
( 49,177)( 50,180)( 51,179)( 52,178)( 53,217)( 54,220)( 55,219)( 56,218)
( 57,213)( 58,216)( 59,215)( 60,214)( 61,209)( 62,212)( 63,211)( 64,210)
( 65,205)( 66,208)( 67,207)( 68,206)( 69,201)( 70,204)( 71,203)( 72,202)
( 73,197)( 74,200)( 75,199)( 76,198)( 77,193)( 78,196)( 79,195)( 80,194)
( 81,189)( 82,192)( 83,191)( 84,190)( 85,185)( 86,188)( 87,187)( 88,186)
( 89,137)( 90,140)( 91,139)( 92,138)( 93,133)( 94,136)( 95,135)( 96,134)
( 97,173)( 98,176)( 99,175)(100,174)(101,169)(102,172)(103,171)(104,170)
(105,165)(106,168)(107,167)(108,166)(109,161)(110,164)(111,163)(112,162)
(113,157)(114,160)(115,159)(116,158)(117,153)(118,156)(119,155)(120,154)
(121,149)(122,152)(123,151)(124,150)(125,145)(126,148)(127,147)(128,146)
(129,141)(130,144)(131,143)(132,142);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope