Polytope of Type {8,66}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,66}*1056
Also Known As : {8,66|2}. if this polytope has another name.
Group : SmallGroup(1056,786)
Rank : 3
Schlafli Type : {8,66}
Number of vertices, edges, etc : 8, 264, 66
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 : {4,66}*528a
   3-fold quotients : {8,22}*352
   4-fold quotients : {2,66}*264
   6-fold quotients : {4,22}*176
   8-fold quotients : {2,33}*132
   11-fold quotients : {8,6}*96
   12-fold quotients : {2,22}*88
   22-fold quotients : {4,6}*48a
   24-fold quotients : {2,11}*44
   33-fold quotients : {8,2}*32
   44-fold quotients : {2,6}*24
   66-fold quotients : {4,2}*16
   88-fold quotients : {2,3}*12
   132-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
to this polytope