Polytope of Type {2,66,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,66,4}*1056c
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}*96b
   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 := (  3,138)(  4,137)(  5,136)(  6,135)(  7,142)(  8,141)(  9,140)( 10,139)
( 11,146)( 12,145)( 13,144)( 14,143)( 15,150)( 16,149)( 17,148)( 18,147)
( 19,154)( 20,153)( 21,152)( 22,151)( 23,158)( 24,157)( 25,156)( 26,155)
( 27,162)( 28,161)( 29,160)( 30,159)( 31,166)( 32,165)( 33,164)( 34,163)
( 35,170)( 36,169)( 37,168)( 38,167)( 39,174)( 40,173)( 41,172)( 42,171)
( 43,178)( 44,177)( 45,176)( 46,175)( 47,182)( 48,181)( 49,180)( 50,179)
( 51,186)( 52,185)( 53,184)( 54,183)( 55,190)( 56,189)( 57,188)( 58,187)
( 59,194)( 60,193)( 61,192)( 62,191)( 63,198)( 64,197)( 65,196)( 66,195)
( 67,202)( 68,201)( 69,200)( 70,199)( 71,206)( 72,205)( 73,204)( 74,203)
( 75,210)( 76,209)( 77,208)( 78,207)( 79,214)( 80,213)( 81,212)( 82,211)
( 83,218)( 84,217)( 85,216)( 86,215)( 87,222)( 88,221)( 89,220)( 90,219)
( 91,226)( 92,225)( 93,224)( 94,223)( 95,230)( 96,229)( 97,228)( 98,227)
( 99,234)(100,233)(101,232)(102,231)(103,238)(104,237)(105,236)(106,235)
(107,242)(108,241)(109,240)(110,239)(111,246)(112,245)(113,244)(114,243)
(115,250)(116,249)(117,248)(118,247)(119,254)(120,253)(121,252)(122,251)
(123,258)(124,257)(125,256)(126,255)(127,262)(128,261)(129,260)(130,259)
(131,266)(132,265)(133,264)(134,263);;
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, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*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 ];;
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)!(  3,138)(  4,137)(  5,136)(  6,135)(  7,142)(  8,141)(  9,140)
( 10,139)( 11,146)( 12,145)( 13,144)( 14,143)( 15,150)( 16,149)( 17,148)
( 18,147)( 19,154)( 20,153)( 21,152)( 22,151)( 23,158)( 24,157)( 25,156)
( 26,155)( 27,162)( 28,161)( 29,160)( 30,159)( 31,166)( 32,165)( 33,164)
( 34,163)( 35,170)( 36,169)( 37,168)( 38,167)( 39,174)( 40,173)( 41,172)
( 42,171)( 43,178)( 44,177)( 45,176)( 46,175)( 47,182)( 48,181)( 49,180)
( 50,179)( 51,186)( 52,185)( 53,184)( 54,183)( 55,190)( 56,189)( 57,188)
( 58,187)( 59,194)( 60,193)( 61,192)( 62,191)( 63,198)( 64,197)( 65,196)
( 66,195)( 67,202)( 68,201)( 69,200)( 70,199)( 71,206)( 72,205)( 73,204)
( 74,203)( 75,210)( 76,209)( 77,208)( 78,207)( 79,214)( 80,213)( 81,212)
( 82,211)( 83,218)( 84,217)( 85,216)( 86,215)( 87,222)( 88,221)( 89,220)
( 90,219)( 91,226)( 92,225)( 93,224)( 94,223)( 95,230)( 96,229)( 97,228)
( 98,227)( 99,234)(100,233)(101,232)(102,231)(103,238)(104,237)(105,236)
(106,235)(107,242)(108,241)(109,240)(110,239)(111,246)(112,245)(113,244)
(114,243)(115,250)(116,249)(117,248)(118,247)(119,254)(120,253)(121,252)
(122,251)(123,258)(124,257)(125,256)(126,255)(127,262)(128,261)(129,260)
(130,259)(131,266)(132,265)(133,264)(134,263);
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, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*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 >;

to this polytope