Polytope of Type {264}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {264}*528
Also Known As : 264-gon, {264}. if this polytope has another name.
Group : SmallGroup(528,66)
Rank : 2
Schlafli Type : {264}
Number of vertices, edges, etc : 264, 264
Order of s₀s₁ : 264
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {264,2} of size 1056
Vertex Figure Of :
   {2,264} of size 1056
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {132}*264
   3-fold quotients : {88}*176
   4-fold quotients : {66}*132
   6-fold quotients : {44}*88
   8-fold quotients : {33}*66
   11-fold quotients : {24}*48
   12-fold quotients : {22}*44
   22-fold quotients : {12}*24
   24-fold quotients : {11}*22
   33-fold quotients : {8}*16
   44-fold quotients : {6}*12
   66-fold quotients : {4}*8
   88-fold quotients : {3}*6
   132-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {528}*1056
   3-fold covers : {792}*1584
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)
( 67,100)( 68,110)( 69,109)( 70,108)( 71,107)( 72,106)( 73,105)( 74,104)
( 75,103)( 76,102)( 77,101)( 78,122)( 79,132)( 80,131)( 81,130)( 82,129)
( 83,128)( 84,127)( 85,126)( 86,125)( 87,124)( 88,123)( 89,111)( 90,121)
( 91,120)( 92,119)( 93,118)( 94,117)( 95,116)( 96,115)( 97,114)( 98,113)
( 99,112)(133,199)(134,209)(135,208)(136,207)(137,206)(138,205)(139,204)
(140,203)(141,202)(142,201)(143,200)(144,221)(145,231)(146,230)(147,229)
(148,228)(149,227)(150,226)(151,225)(152,224)(153,223)(154,222)(155,210)
(156,220)(157,219)(158,218)(159,217)(160,216)(161,215)(162,214)(163,213)
(164,212)(165,211)(166,232)(167,242)(168,241)(169,240)(170,239)(171,238)
(172,237)(173,236)(174,235)(175,234)(176,233)(177,254)(178,264)(179,263)
(180,262)(181,261)(182,260)(183,259)(184,258)(185,257)(186,256)(187,255)
(188,243)(189,253)(190,252)(191,251)(192,250)(193,249)(194,248)(195,247)
(196,246)(197,245)(198,244);;
s1 := (  1,145)(  2,144)(  3,154)(  4,153)(  5,152)(  6,151)(  7,150)(  8,149)
(  9,148)( 10,147)( 11,146)( 12,134)( 13,133)( 14,143)( 15,142)( 16,141)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,136)( 22,135)( 23,156)( 24,155)
( 25,165)( 26,164)( 27,163)( 28,162)( 29,161)( 30,160)( 31,159)( 32,158)
( 33,157)( 34,178)( 35,177)( 36,187)( 37,186)( 38,185)( 39,184)( 40,183)
( 41,182)( 42,181)( 43,180)( 44,179)( 45,167)( 46,166)( 47,176)( 48,175)
( 49,174)( 50,173)( 51,172)( 52,171)( 53,170)( 54,169)( 55,168)( 56,189)
( 57,188)( 58,198)( 59,197)( 60,196)( 61,195)( 62,194)( 63,193)( 64,192)
( 65,191)( 66,190)( 67,244)( 68,243)( 69,253)( 70,252)( 71,251)( 72,250)
( 73,249)( 74,248)( 75,247)( 76,246)( 77,245)( 78,233)( 79,232)( 80,242)
( 81,241)( 82,240)( 83,239)( 84,238)( 85,237)( 86,236)( 87,235)( 88,234)
( 89,255)( 90,254)( 91,264)( 92,263)( 93,262)( 94,261)( 95,260)( 96,259)
( 97,258)( 98,257)( 99,256)(100,211)(101,210)(102,220)(103,219)(104,218)
(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,200)(112,199)
(113,209)(114,208)(115,207)(116,206)(117,205)(118,204)(119,203)(120,202)
(121,201)(122,222)(123,221)(124,231)(125,230)(126,229)(127,228)(128,227)
(129,226)(130,225)(131,224)(132,223);;
poly := Group([s0,s1]);;

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1> := Group< s0,s1 | s0*s0, s1*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*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*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*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*s0*s1 >;

References : None.
to this polytope