Polytope of Type {2,66,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,66,4}*1056b
if this polytope has a name.
Group : SmallGroup(1056,1017)
Rank : 4
Schlafli Type : {2,66,4}
Number of vertices, edges, etc : 2, 66, 132, 4
Order of s₀s₁s₂s₃ : 66
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
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 : {2,33,4}*528
   11-fold quotients : {2,6,4}*96c
   22-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7, 43)(  8, 45)(  9, 44)( 10, 46)( 11, 39)( 12, 41)( 13, 40)
( 14, 42)( 15, 35)( 16, 37)( 17, 36)( 18, 38)( 19, 31)( 20, 33)( 21, 32)
( 22, 34)( 23, 27)( 24, 29)( 25, 28)( 26, 30)( 47, 91)( 48, 93)( 49, 92)
( 50, 94)( 51,131)( 52,133)( 53,132)( 54,134)( 55,127)( 56,129)( 57,128)
( 58,130)( 59,123)( 60,125)( 61,124)( 62,126)( 63,119)( 64,121)( 65,120)
( 66,122)( 67,115)( 68,117)( 69,116)( 70,118)( 71,111)( 72,113)( 73,112)
( 74,114)( 75,107)( 76,109)( 77,108)( 78,110)( 79,103)( 80,105)( 81,104)
( 82,106)( 83, 99)( 84,101)( 85,100)( 86,102)( 87, 95)( 88, 97)( 89, 96)
( 90, 98)(136,137)(139,175)(140,177)(141,176)(142,178)(143,171)(144,173)
(145,172)(146,174)(147,167)(148,169)(149,168)(150,170)(151,163)(152,165)
(153,164)(154,166)(155,159)(156,161)(157,160)(158,162)(179,223)(180,225)
(181,224)(182,226)(183,263)(184,265)(185,264)(186,266)(187,259)(188,261)
(189,260)(190,262)(191,255)(192,257)(193,256)(194,258)(195,251)(196,253)
(197,252)(198,254)(199,247)(200,249)(201,248)(202,250)(203,243)(204,245)
(205,244)(206,246)(207,239)(208,241)(209,240)(210,242)(211,235)(212,237)
(213,236)(214,238)(215,231)(216,233)(217,232)(218,234)(219,227)(220,229)
(221,228)(222,230);;
s2 := (  3,183)(  4,184)(  5,186)(  6,185)(  7,179)(  8,180)(  9,182)( 10,181)
( 11,219)( 12,220)( 13,222)( 14,221)( 15,215)( 16,216)( 17,218)( 18,217)
( 19,211)( 20,212)( 21,214)( 22,213)( 23,207)( 24,208)( 25,210)( 26,209)
( 27,203)( 28,204)( 29,206)( 30,205)( 31,199)( 32,200)( 33,202)( 34,201)
( 35,195)( 36,196)( 37,198)( 38,197)( 39,191)( 40,192)( 41,194)( 42,193)
( 43,187)( 44,188)( 45,190)( 46,189)( 47,139)( 48,140)( 49,142)( 50,141)
( 51,135)( 52,136)( 53,138)( 54,137)( 55,175)( 56,176)( 57,178)( 58,177)
( 59,171)( 60,172)( 61,174)( 62,173)( 63,167)( 64,168)( 65,170)( 66,169)
( 67,163)( 68,164)( 69,166)( 70,165)( 71,159)( 72,160)( 73,162)( 74,161)
( 75,155)( 76,156)( 77,158)( 78,157)( 79,151)( 80,152)( 81,154)( 82,153)
( 83,147)( 84,148)( 85,150)( 86,149)( 87,143)( 88,144)( 89,146)( 90,145)
( 91,227)( 92,228)( 93,230)( 94,229)( 95,223)( 96,224)( 97,226)( 98,225)
( 99,263)(100,264)(101,266)(102,265)(103,259)(104,260)(105,262)(106,261)
(107,255)(108,256)(109,258)(110,257)(111,251)(112,252)(113,254)(114,253)
(115,247)(116,248)(117,250)(118,249)(119,243)(120,244)(121,246)(122,245)
(123,239)(124,240)(125,242)(126,241)(127,235)(128,236)(129,238)(130,237)
(131,231)(132,232)(133,234)(134,233);;
s3 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)
( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)
( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)
( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)
( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)(112,113)
(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)(128,129)
(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)(144,145)
(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)(160,161)
(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)(176,177)
(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)
(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)(208,209)
(211,214)(212,213)(215,218)(216,217)(219,222)(220,221)(223,226)(224,225)
(227,230)(228,229)(231,234)(232,233)(235,238)(236,237)(239,242)(240,241)
(243,246)(244,245)(247,250)(248,249)(251,254)(252,253)(255,258)(256,257)
(259,262)(260,261)(263,266)(264,265);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*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*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(266)!(1,2);
s1 := Sym(266)!(  4,  5)(  7, 43)(  8, 45)(  9, 44)( 10, 46)( 11, 39)( 12, 41)
( 13, 40)( 14, 42)( 15, 35)( 16, 37)( 17, 36)( 18, 38)( 19, 31)( 20, 33)
( 21, 32)( 22, 34)( 23, 27)( 24, 29)( 25, 28)( 26, 30)( 47, 91)( 48, 93)
( 49, 92)( 50, 94)( 51,131)( 52,133)( 53,132)( 54,134)( 55,127)( 56,129)
( 57,128)( 58,130)( 59,123)( 60,125)( 61,124)( 62,126)( 63,119)( 64,121)
( 65,120)( 66,122)( 67,115)( 68,117)( 69,116)( 70,118)( 71,111)( 72,113)
( 73,112)( 74,114)( 75,107)( 76,109)( 77,108)( 78,110)( 79,103)( 80,105)
( 81,104)( 82,106)( 83, 99)( 84,101)( 85,100)( 86,102)( 87, 95)( 88, 97)
( 89, 96)( 90, 98)(136,137)(139,175)(140,177)(141,176)(142,178)(143,171)
(144,173)(145,172)(146,174)(147,167)(148,169)(149,168)(150,170)(151,163)
(152,165)(153,164)(154,166)(155,159)(156,161)(157,160)(158,162)(179,223)
(180,225)(181,224)(182,226)(183,263)(184,265)(185,264)(186,266)(187,259)
(188,261)(189,260)(190,262)(191,255)(192,257)(193,256)(194,258)(195,251)
(196,253)(197,252)(198,254)(199,247)(200,249)(201,248)(202,250)(203,243)
(204,245)(205,244)(206,246)(207,239)(208,241)(209,240)(210,242)(211,235)
(212,237)(213,236)(214,238)(215,231)(216,233)(217,232)(218,234)(219,227)
(220,229)(221,228)(222,230);
s2 := Sym(266)!(  3,183)(  4,184)(  5,186)(  6,185)(  7,179)(  8,180)(  9,182)
( 10,181)( 11,219)( 12,220)( 13,222)( 14,221)( 15,215)( 16,216)( 17,218)
( 18,217)( 19,211)( 20,212)( 21,214)( 22,213)( 23,207)( 24,208)( 25,210)
( 26,209)( 27,203)( 28,204)( 29,206)( 30,205)( 31,199)( 32,200)( 33,202)
( 34,201)( 35,195)( 36,196)( 37,198)( 38,197)( 39,191)( 40,192)( 41,194)
( 42,193)( 43,187)( 44,188)( 45,190)( 46,189)( 47,139)( 48,140)( 49,142)
( 50,141)( 51,135)( 52,136)( 53,138)( 54,137)( 55,175)( 56,176)( 57,178)
( 58,177)( 59,171)( 60,172)( 61,174)( 62,173)( 63,167)( 64,168)( 65,170)
( 66,169)( 67,163)( 68,164)( 69,166)( 70,165)( 71,159)( 72,160)( 73,162)
( 74,161)( 75,155)( 76,156)( 77,158)( 78,157)( 79,151)( 80,152)( 81,154)
( 82,153)( 83,147)( 84,148)( 85,150)( 86,149)( 87,143)( 88,144)( 89,146)
( 90,145)( 91,227)( 92,228)( 93,230)( 94,229)( 95,223)( 96,224)( 97,226)
( 98,225)( 99,263)(100,264)(101,266)(102,265)(103,259)(104,260)(105,262)
(106,261)(107,255)(108,256)(109,258)(110,257)(111,251)(112,252)(113,254)
(114,253)(115,247)(116,248)(117,250)(118,249)(119,243)(120,244)(121,246)
(122,245)(123,239)(124,240)(125,242)(126,241)(127,235)(128,236)(129,238)
(130,237)(131,231)(132,232)(133,234)(134,233);
s3 := Sym(266)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)
( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)
( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)
( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)
( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)
(112,113)(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)
(128,129)(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)
(144,145)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)
(160,161)(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)
(176,177)(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)
(192,193)(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)
(208,209)(211,214)(212,213)(215,218)(216,217)(219,222)(220,221)(223,226)
(224,225)(227,230)(228,229)(231,234)(232,233)(235,238)(236,237)(239,242)
(240,241)(243,246)(244,245)(247,250)(248,249)(251,254)(252,253)(255,258)
(256,257)(259,262)(260,261)(263,266)(264,265);
poly := sub<Sym(266)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*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*s1*s2*s1*s2 >;

to this polytope