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Polytope of Type {66,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {66,8}*1056
Also Known As : {66,8|2}. if this polytope has another name.
Group : SmallGroup(1056,786)
Rank : 3
Schlafli Type : {66,8}
Number of vertices, edges, etc : 66, 264, 8
Order of s0s1s2 : 264
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {66,4}*528a
   3-fold quotients : {22,8}*352
   4-fold quotients : {66,2}*264
   6-fold quotients : {22,4}*176
   8-fold quotients : {33,2}*132
   11-fold quotients : {6,8}*96
   12-fold quotients : {22,2}*88
   22-fold quotients : {6,4}*48a
   24-fold quotients : {11,2}*44
   33-fold quotients : {2,8}*32
   44-fold quotients : {6,2}*24
   66-fold quotients : {2,4}*16
   88-fold quotients : {3,2}*12
   132-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)( 14, 32)
( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)
( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 45, 56)( 46, 66)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)( 55, 57)
( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 78, 89)( 79, 99)( 80, 98)
( 81, 97)( 82, 96)( 83, 95)( 84, 94)( 85, 93)( 86, 92)( 87, 91)( 88, 90)
(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)(113,131)
(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)
(134,143)(135,142)(136,141)(137,140)(138,139)(144,155)(145,165)(146,164)
(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)
(167,176)(168,175)(169,174)(170,173)(171,172)(177,188)(178,198)(179,197)
(180,196)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)
(200,209)(201,208)(202,207)(203,206)(204,205)(210,221)(211,231)(212,230)
(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222)
(233,242)(234,241)(235,240)(236,239)(237,238)(243,254)(244,264)(245,263)
(246,262)(247,261)(248,260)(249,259)(250,258)(251,257)(252,256)(253,255);;
s1 := (  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)(  8, 17)
(  9, 16)( 10, 15)( 11, 14)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)
( 34, 46)( 35, 45)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)
( 42, 49)( 43, 48)( 44, 47)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)
( 67,112)( 68,111)( 69,121)( 70,120)( 71,119)( 72,118)( 73,117)( 74,116)
( 75,115)( 76,114)( 77,113)( 78,101)( 79,100)( 80,110)( 81,109)( 82,108)
( 83,107)( 84,106)( 85,105)( 86,104)( 87,103)( 88,102)( 89,123)( 90,122)
( 91,132)( 92,131)( 93,130)( 94,129)( 95,128)( 96,127)( 97,126)( 98,125)
( 99,124)(133,211)(134,210)(135,220)(136,219)(137,218)(138,217)(139,216)
(140,215)(141,214)(142,213)(143,212)(144,200)(145,199)(146,209)(147,208)
(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,222)
(156,221)(157,231)(158,230)(159,229)(160,228)(161,227)(162,226)(163,225)
(164,224)(165,223)(166,244)(167,243)(168,253)(169,252)(170,251)(171,250)
(172,249)(173,248)(174,247)(175,246)(176,245)(177,233)(178,232)(179,242)
(180,241)(181,240)(182,239)(183,238)(184,237)(185,236)(186,235)(187,234)
(188,255)(189,254)(190,264)(191,263)(192,262)(193,261)(194,260)(195,259)
(196,258)(197,257)(198,256);;
s2 := (  1,133)(  2,134)(  3,135)(  4,136)(  5,137)(  6,138)(  7,139)(  8,140)
(  9,141)( 10,142)( 11,143)( 12,144)( 13,145)( 14,146)( 15,147)( 16,148)
( 17,149)( 18,150)( 19,151)( 20,152)( 21,153)( 22,154)( 23,155)( 24,156)
( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)( 32,164)
( 33,165)( 34,166)( 35,167)( 36,168)( 37,169)( 38,170)( 39,171)( 40,172)
( 41,173)( 42,174)( 43,175)( 44,176)( 45,177)( 46,178)( 47,179)( 48,180)
( 49,181)( 50,182)( 51,183)( 52,184)( 53,185)( 54,186)( 55,187)( 56,188)
( 57,189)( 58,190)( 59,191)( 60,192)( 61,193)( 62,194)( 63,195)( 64,196)
( 65,197)( 66,198)( 67,232)( 68,233)( 69,234)( 70,235)( 71,236)( 72,237)
( 73,238)( 74,239)( 75,240)( 76,241)( 77,242)( 78,243)( 79,244)( 80,245)
( 81,246)( 82,247)( 83,248)( 84,249)( 85,250)( 86,251)( 87,252)( 88,253)
( 89,254)( 90,255)( 91,256)( 92,257)( 93,258)( 94,259)( 95,260)( 96,261)
( 97,262)( 98,263)( 99,264)(100,199)(101,200)(102,201)(103,202)(104,203)
(105,204)(106,205)(107,206)(108,207)(109,208)(110,209)(111,210)(112,211)
(113,212)(114,213)(115,214)(116,215)(117,216)(118,217)(119,218)(120,219)
(121,220)(122,221)(123,222)(124,223)(125,224)(126,225)(127,226)(128,227)
(129,228)(130,229)(131,230)(132,231);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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(264)!(  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)
( 14, 32)( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)
( 22, 24)( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 45, 56)( 46, 66)
( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)
( 55, 57)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 78, 89)( 79, 99)
( 80, 98)( 81, 97)( 82, 96)( 83, 95)( 84, 94)( 85, 93)( 86, 92)( 87, 91)
( 88, 90)(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)
(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)
(121,123)(134,143)(135,142)(136,141)(137,140)(138,139)(144,155)(145,165)
(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)
(154,156)(167,176)(168,175)(169,174)(170,173)(171,172)(177,188)(178,198)
(179,197)(180,196)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)
(187,189)(200,209)(201,208)(202,207)(203,206)(204,205)(210,221)(211,231)
(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)
(220,222)(233,242)(234,241)(235,240)(236,239)(237,238)(243,254)(244,264)
(245,263)(246,262)(247,261)(248,260)(249,259)(250,258)(251,257)(252,256)
(253,255);
s1 := Sym(264)!(  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)
(  8, 17)(  9, 16)( 10, 15)( 11, 14)( 23, 24)( 25, 33)( 26, 32)( 27, 31)
( 28, 30)( 34, 46)( 35, 45)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)
( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 56, 57)( 58, 66)( 59, 65)( 60, 64)
( 61, 63)( 67,112)( 68,111)( 69,121)( 70,120)( 71,119)( 72,118)( 73,117)
( 74,116)( 75,115)( 76,114)( 77,113)( 78,101)( 79,100)( 80,110)( 81,109)
( 82,108)( 83,107)( 84,106)( 85,105)( 86,104)( 87,103)( 88,102)( 89,123)
( 90,122)( 91,132)( 92,131)( 93,130)( 94,129)( 95,128)( 96,127)( 97,126)
( 98,125)( 99,124)(133,211)(134,210)(135,220)(136,219)(137,218)(138,217)
(139,216)(140,215)(141,214)(142,213)(143,212)(144,200)(145,199)(146,209)
(147,208)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)
(155,222)(156,221)(157,231)(158,230)(159,229)(160,228)(161,227)(162,226)
(163,225)(164,224)(165,223)(166,244)(167,243)(168,253)(169,252)(170,251)
(171,250)(172,249)(173,248)(174,247)(175,246)(176,245)(177,233)(178,232)
(179,242)(180,241)(181,240)(182,239)(183,238)(184,237)(185,236)(186,235)
(187,234)(188,255)(189,254)(190,264)(191,263)(192,262)(193,261)(194,260)
(195,259)(196,258)(197,257)(198,256);
s2 := Sym(264)!(  1,133)(  2,134)(  3,135)(  4,136)(  5,137)(  6,138)(  7,139)
(  8,140)(  9,141)( 10,142)( 11,143)( 12,144)( 13,145)( 14,146)( 15,147)
( 16,148)( 17,149)( 18,150)( 19,151)( 20,152)( 21,153)( 22,154)( 23,155)
( 24,156)( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)
( 32,164)( 33,165)( 34,166)( 35,167)( 36,168)( 37,169)( 38,170)( 39,171)
( 40,172)( 41,173)( 42,174)( 43,175)( 44,176)( 45,177)( 46,178)( 47,179)
( 48,180)( 49,181)( 50,182)( 51,183)( 52,184)( 53,185)( 54,186)( 55,187)
( 56,188)( 57,189)( 58,190)( 59,191)( 60,192)( 61,193)( 62,194)( 63,195)
( 64,196)( 65,197)( 66,198)( 67,232)( 68,233)( 69,234)( 70,235)( 71,236)
( 72,237)( 73,238)( 74,239)( 75,240)( 76,241)( 77,242)( 78,243)( 79,244)
( 80,245)( 81,246)( 82,247)( 83,248)( 84,249)( 85,250)( 86,251)( 87,252)
( 88,253)( 89,254)( 90,255)( 91,256)( 92,257)( 93,258)( 94,259)( 95,260)
( 96,261)( 97,262)( 98,263)( 99,264)(100,199)(101,200)(102,201)(103,202)
(104,203)(105,204)(106,205)(107,206)(108,207)(109,208)(110,209)(111,210)
(112,211)(113,212)(114,213)(115,214)(116,215)(117,216)(118,217)(119,218)
(120,219)(121,220)(122,221)(123,222)(124,223)(125,224)(126,225)(127,226)
(128,227)(129,228)(130,229)(131,230)(132,231);
poly := sub<Sym(264)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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