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