Polytope of Type {6,66}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,66}*1056
if this polytope has a name.
Group : SmallGroup(1056,1015)
Rank : 3
Schlafli Type : {6,66}
Number of vertices, edges, etc : 8, 264, 88
Order of s₀s₁s₂ : 44
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,33}*528
   11-fold quotients : {6,6}*96
   12-fold quotients : {2,22}*88
   22-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {2,11}*44
   44-fold quotients : {3,3}*24
   132-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope