Polytope of Type {2,4,66}

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

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

Permutation Representation (Magma) :

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

to this polytope