Polytope of Type {4,33,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,33,2}*1056
if this polytope has a name.
Group : SmallGroup(1056,1017)
Rank : 4
Schlafli Type : {4,33,2}
Number of vertices, edges, etc : 8, 132, 66, 2
Order of s₀s₁s₂s₃ : 66
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {4,33,2}*528
   4-fold quotients : {2,33,2}*264
   11-fold quotients : {4,3,2}*96
   12-fold quotients : {2,11,2}*88
   22-fold quotients : {4,3,2}*48
   44-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 93)(  2, 96)(  3, 95)(  4, 94)(  5, 89)(  6, 92)(  7, 91)(  8, 90)
(  9,129)( 10,132)( 11,131)( 12,130)( 13,125)( 14,128)( 15,127)( 16,126)
( 17,121)( 18,124)( 19,123)( 20,122)( 21,117)( 22,120)( 23,119)( 24,118)
( 25,113)( 26,116)( 27,115)( 28,114)( 29,109)( 30,112)( 31,111)( 32,110)
( 33,105)( 34,108)( 35,107)( 36,106)( 37,101)( 38,104)( 39,103)( 40,102)
( 41, 97)( 42,100)( 43, 99)( 44, 98)( 45, 49)( 46, 52)( 47, 51)( 48, 50)
( 53, 85)( 54, 88)( 55, 87)( 56, 86)( 57, 81)( 58, 84)( 59, 83)( 60, 82)
( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 65, 73)( 66, 76)( 67, 75)( 68, 74)
( 70, 72)(133,225)(134,228)(135,227)(136,226)(137,221)(138,224)(139,223)
(140,222)(141,261)(142,264)(143,263)(144,262)(145,257)(146,260)(147,259)
(148,258)(149,253)(150,256)(151,255)(152,254)(153,249)(154,252)(155,251)
(156,250)(157,245)(158,248)(159,247)(160,246)(161,241)(162,244)(163,243)
(164,242)(165,237)(166,240)(167,239)(168,238)(169,233)(170,236)(171,235)
(172,234)(173,229)(174,232)(175,231)(176,230)(177,181)(178,184)(179,183)
(180,182)(185,217)(186,220)(187,219)(188,218)(189,213)(190,216)(191,215)
(192,214)(193,209)(194,212)(195,211)(196,210)(197,205)(198,208)(199,207)
(200,206)(202,204);;
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, 
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*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)!(  1,135)(  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(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);
s2 := Sym(266)!(  1, 93)(  2, 96)(  3, 95)(  4, 94)(  5, 89)(  6, 92)(  7, 91)
(  8, 90)(  9,129)( 10,132)( 11,131)( 12,130)( 13,125)( 14,128)( 15,127)
( 16,126)( 17,121)( 18,124)( 19,123)( 20,122)( 21,117)( 22,120)( 23,119)
( 24,118)( 25,113)( 26,116)( 27,115)( 28,114)( 29,109)( 30,112)( 31,111)
( 32,110)( 33,105)( 34,108)( 35,107)( 36,106)( 37,101)( 38,104)( 39,103)
( 40,102)( 41, 97)( 42,100)( 43, 99)( 44, 98)( 45, 49)( 46, 52)( 47, 51)
( 48, 50)( 53, 85)( 54, 88)( 55, 87)( 56, 86)( 57, 81)( 58, 84)( 59, 83)
( 60, 82)( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 65, 73)( 66, 76)( 67, 75)
( 68, 74)( 70, 72)(133,225)(134,228)(135,227)(136,226)(137,221)(138,224)
(139,223)(140,222)(141,261)(142,264)(143,263)(144,262)(145,257)(146,260)
(147,259)(148,258)(149,253)(150,256)(151,255)(152,254)(153,249)(154,252)
(155,251)(156,250)(157,245)(158,248)(159,247)(160,246)(161,241)(162,244)
(163,243)(164,242)(165,237)(166,240)(167,239)(168,238)(169,233)(170,236)
(171,235)(172,234)(173,229)(174,232)(175,231)(176,230)(177,181)(178,184)
(179,183)(180,182)(185,217)(186,220)(187,219)(188,218)(189,213)(190,216)
(191,215)(192,214)(193,209)(194,212)(195,211)(196,210)(197,205)(198,208)
(199,207)(200,206)(202,204);
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, 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*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