Polytope of Type {66,4,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {66,4,2}*1056c
if this polytope has a name.
Group : SmallGroup(1056,1017)
Rank : 4
Schlafli Type : {66,4,2}
Number of vertices, edges, etc : 66, 132, 4, 2
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 : {33,4,2}*528
   11-fold quotients : {6,4,2}*96b
   22-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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 := (  1,181)(  2,182)(  3,184)(  4,183)(  5,177)(  6,178)(  7,180)(  8,179)
(  9,217)( 10,218)( 11,220)( 12,219)( 13,213)( 14,214)( 15,216)( 16,215)
( 17,209)( 18,210)( 19,212)( 20,211)( 21,205)( 22,206)( 23,208)( 24,207)
( 25,201)( 26,202)( 27,204)( 28,203)( 29,197)( 30,198)( 31,200)( 32,199)
( 33,193)( 34,194)( 35,196)( 36,195)( 37,189)( 38,190)( 39,192)( 40,191)
( 41,185)( 42,186)( 43,188)( 44,187)( 45,137)( 46,138)( 47,140)( 48,139)
( 49,133)( 50,134)( 51,136)( 52,135)( 53,173)( 54,174)( 55,176)( 56,175)
( 57,169)( 58,170)( 59,172)( 60,171)( 61,165)( 62,166)( 63,168)( 64,167)
( 65,161)( 66,162)( 67,164)( 68,163)( 69,157)( 70,158)( 71,160)( 72,159)
( 73,153)( 74,154)( 75,156)( 76,155)( 77,149)( 78,150)( 79,152)( 80,151)
( 81,145)( 82,146)( 83,148)( 84,147)( 85,141)( 86,142)( 87,144)( 88,143)
( 89,225)( 90,226)( 91,228)( 92,227)( 93,221)( 94,222)( 95,224)( 96,223)
( 97,261)( 98,262)( 99,264)(100,263)(101,257)(102,258)(103,260)(104,259)
(105,253)(106,254)(107,256)(108,255)(109,249)(110,250)(111,252)(112,251)
(113,245)(114,246)(115,248)(116,247)(117,241)(118,242)(119,244)(120,243)
(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)(127,236)(128,235)
(129,229)(130,230)(131,232)(132,231);;
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);;
s3 := (265,266);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
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*s2*s1*s0*s2*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope