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Polytope of Type {33,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {33,4}*528
if this polytope has a name.
Group : SmallGroup(528,162)
Rank : 3
Schlafli Type : {33,4}
Number of vertices, edges, etc : 66, 132, 8
Order of s0s1s2 : 66
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {33,4,2} of size 1056
Vertex Figure Of :
   {2,33,4} of size 1056
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {33,4}*264
   4-fold quotients : {33,2}*132
   11-fold quotients : {3,4}*48
   12-fold quotients : {11,2}*44
   22-fold quotients : {3,4}*24
   44-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {33,8}*1056, {66,4}*1056
   3-fold covers : {99,4}*1584, {33,12}*1584
Permutation Representation (GAP) :
s0 := (  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);;
s1 := (  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 45)(  6, 46)(  7, 48)(  8, 47)
(  9, 85)( 10, 86)( 11, 88)( 12, 87)( 13, 81)( 14, 82)( 15, 84)( 16, 83)
( 17, 77)( 18, 78)( 19, 80)( 20, 79)( 21, 73)( 22, 74)( 23, 76)( 24, 75)
( 25, 69)( 26, 70)( 27, 72)( 28, 71)( 29, 65)( 30, 66)( 31, 68)( 32, 67)
( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 37, 57)( 38, 58)( 39, 60)( 40, 59)
( 41, 53)( 42, 54)( 43, 56)( 44, 55)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 97,129)( 98,130)( 99,132)(100,131)(101,125)(102,126)(103,128)(104,127)
(105,121)(106,122)(107,124)(108,123)(109,117)(110,118)(111,120)(112,119)
(115,116)(133,181)(134,182)(135,184)(136,183)(137,177)(138,178)(139,180)
(140,179)(141,217)(142,218)(143,220)(144,219)(145,213)(146,214)(147,216)
(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)(155,208)
(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)(163,200)
(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)(170,190)(171,192)
(172,191)(173,185)(174,186)(175,188)(176,187)(221,225)(222,226)(223,228)
(224,227)(229,261)(230,262)(231,264)(232,263)(233,257)(234,258)(235,260)
(236,259)(237,253)(238,254)(239,256)(240,255)(241,249)(242,250)(243,252)
(244,251)(247,248);;
s2 := (  1,136)(  2,135)(  3,134)(  4,133)(  5,140)(  6,139)(  7,138)(  8,137)
(  9,144)( 10,143)( 11,142)( 12,141)( 13,148)( 14,147)( 15,146)( 16,145)
( 17,152)( 18,151)( 19,150)( 20,149)( 21,156)( 22,155)( 23,154)( 24,153)
( 25,160)( 26,159)( 27,158)( 28,157)( 29,164)( 30,163)( 31,162)( 32,161)
( 33,168)( 34,167)( 35,166)( 36,165)( 37,172)( 38,171)( 39,170)( 40,169)
( 41,176)( 42,175)( 43,174)( 44,173)( 45,180)( 46,179)( 47,178)( 48,177)
( 49,184)( 50,183)( 51,182)( 52,181)( 53,188)( 54,187)( 55,186)( 56,185)
( 57,192)( 58,191)( 59,190)( 60,189)( 61,196)( 62,195)( 63,194)( 64,193)
( 65,200)( 66,199)( 67,198)( 68,197)( 69,204)( 70,203)( 71,202)( 72,201)
( 73,208)( 74,207)( 75,206)( 76,205)( 77,212)( 78,211)( 79,210)( 80,209)
( 81,216)( 82,215)( 83,214)( 84,213)( 85,220)( 86,219)( 87,218)( 88,217)
( 89,224)( 90,223)( 91,222)( 92,221)( 93,228)( 94,227)( 95,226)( 96,225)
( 97,232)( 98,231)( 99,230)(100,229)(101,236)(102,235)(103,234)(104,233)
(105,240)(106,239)(107,238)(108,237)(109,244)(110,243)(111,242)(112,241)
(113,248)(114,247)(115,246)(116,245)(117,252)(118,251)(119,250)(120,249)
(121,256)(122,255)(123,254)(124,253)(125,260)(126,259)(127,258)(128,257)
(129,264)(130,263)(131,262)(132,261);;
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*s1*s2*s1*s0*s1*s0*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(264)!(  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 45)(  6, 46)(  7, 48)
(  8, 47)(  9, 85)( 10, 86)( 11, 88)( 12, 87)( 13, 81)( 14, 82)( 15, 84)
( 16, 83)( 17, 77)( 18, 78)( 19, 80)( 20, 79)( 21, 73)( 22, 74)( 23, 76)
( 24, 75)( 25, 69)( 26, 70)( 27, 72)( 28, 71)( 29, 65)( 30, 66)( 31, 68)
( 32, 67)( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 37, 57)( 38, 58)( 39, 60)
( 40, 59)( 41, 53)( 42, 54)( 43, 56)( 44, 55)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 97,129)( 98,130)( 99,132)(100,131)(101,125)(102,126)(103,128)
(104,127)(105,121)(106,122)(107,124)(108,123)(109,117)(110,118)(111,120)
(112,119)(115,116)(133,181)(134,182)(135,184)(136,183)(137,177)(138,178)
(139,180)(140,179)(141,217)(142,218)(143,220)(144,219)(145,213)(146,214)
(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)
(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)
(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)(170,190)
(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(221,225)(222,226)
(223,228)(224,227)(229,261)(230,262)(231,264)(232,263)(233,257)(234,258)
(235,260)(236,259)(237,253)(238,254)(239,256)(240,255)(241,249)(242,250)
(243,252)(244,251)(247,248);
s2 := Sym(264)!(  1,136)(  2,135)(  3,134)(  4,133)(  5,140)(  6,139)(  7,138)
(  8,137)(  9,144)( 10,143)( 11,142)( 12,141)( 13,148)( 14,147)( 15,146)
( 16,145)( 17,152)( 18,151)( 19,150)( 20,149)( 21,156)( 22,155)( 23,154)
( 24,153)( 25,160)( 26,159)( 27,158)( 28,157)( 29,164)( 30,163)( 31,162)
( 32,161)( 33,168)( 34,167)( 35,166)( 36,165)( 37,172)( 38,171)( 39,170)
( 40,169)( 41,176)( 42,175)( 43,174)( 44,173)( 45,180)( 46,179)( 47,178)
( 48,177)( 49,184)( 50,183)( 51,182)( 52,181)( 53,188)( 54,187)( 55,186)
( 56,185)( 57,192)( 58,191)( 59,190)( 60,189)( 61,196)( 62,195)( 63,194)
( 64,193)( 65,200)( 66,199)( 67,198)( 68,197)( 69,204)( 70,203)( 71,202)
( 72,201)( 73,208)( 74,207)( 75,206)( 76,205)( 77,212)( 78,211)( 79,210)
( 80,209)( 81,216)( 82,215)( 83,214)( 84,213)( 85,220)( 86,219)( 87,218)
( 88,217)( 89,224)( 90,223)( 91,222)( 92,221)( 93,228)( 94,227)( 95,226)
( 96,225)( 97,232)( 98,231)( 99,230)(100,229)(101,236)(102,235)(103,234)
(104,233)(105,240)(106,239)(107,238)(108,237)(109,244)(110,243)(111,242)
(112,241)(113,248)(114,247)(115,246)(116,245)(117,252)(118,251)(119,250)
(120,249)(121,256)(122,255)(123,254)(124,253)(125,260)(126,259)(127,258)
(128,257)(129,264)(130,263)(131,262)(132,261);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*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 >; 
 
References : None.
to this polytope