Polytope of Type {268}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {268}*536
Also Known As : 268-gon, {268}. if this polytope has another name.
Group : SmallGroup(536,5)
Rank : 2
Schlafli Type : {268}
Number of vertices, edges, etc : 268, 268
Order of s₀s₁ : 268
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {268,2} of size 1072
Vertex Figure Of :
   {2,268} of size 1072
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {134}*268
   4-fold quotients : {67}*134
   67-fold quotients : {4}*8
   134-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {536}*1072
   3-fold covers : {804}*1608
Permutation Representation (GAP) :

s0 := (  2, 67)(  3, 66)(  4, 65)(  5, 64)(  6, 63)(  7, 62)(  8, 61)(  9, 60)
( 10, 59)( 11, 58)( 12, 57)( 13, 56)( 14, 55)( 15, 54)( 16, 53)( 17, 52)
( 18, 51)( 19, 50)( 20, 49)( 21, 48)( 22, 47)( 23, 46)( 24, 45)( 25, 44)
( 26, 43)( 27, 42)( 28, 41)( 29, 40)( 30, 39)( 31, 38)( 32, 37)( 33, 36)
( 34, 35)( 69,134)( 70,133)( 71,132)( 72,131)( 73,130)( 74,129)( 75,128)
( 76,127)( 77,126)( 78,125)( 79,124)( 80,123)( 81,122)( 82,121)( 83,120)
( 84,119)( 85,118)( 86,117)( 87,116)( 88,115)( 89,114)( 90,113)( 91,112)
( 92,111)( 93,110)( 94,109)( 95,108)( 96,107)( 97,106)( 98,105)( 99,104)
(100,103)(101,102)(135,202)(136,268)(137,267)(138,266)(139,265)(140,264)
(141,263)(142,262)(143,261)(144,260)(145,259)(146,258)(147,257)(148,256)
(149,255)(150,254)(151,253)(152,252)(153,251)(154,250)(155,249)(156,248)
(157,247)(158,246)(159,245)(160,244)(161,243)(162,242)(163,241)(164,240)
(165,239)(166,238)(167,237)(168,236)(169,235)(170,234)(171,233)(172,232)
(173,231)(174,230)(175,229)(176,228)(177,227)(178,226)(179,225)(180,224)
(181,223)(182,222)(183,221)(184,220)(185,219)(186,218)(187,217)(188,216)
(189,215)(190,214)(191,213)(192,212)(193,211)(194,210)(195,209)(196,208)
(197,207)(198,206)(199,205)(200,204)(201,203);;
s1 := (  1,136)(  2,135)(  3,201)(  4,200)(  5,199)(  6,198)(  7,197)(  8,196)
(  9,195)( 10,194)( 11,193)( 12,192)( 13,191)( 14,190)( 15,189)( 16,188)
( 17,187)( 18,186)( 19,185)( 20,184)( 21,183)( 22,182)( 23,181)( 24,180)
( 25,179)( 26,178)( 27,177)( 28,176)( 29,175)( 30,174)( 31,173)( 32,172)
( 33,171)( 34,170)( 35,169)( 36,168)( 37,167)( 38,166)( 39,165)( 40,164)
( 41,163)( 42,162)( 43,161)( 44,160)( 45,159)( 46,158)( 47,157)( 48,156)
( 49,155)( 50,154)( 51,153)( 52,152)( 53,151)( 54,150)( 55,149)( 56,148)
( 57,147)( 58,146)( 59,145)( 60,144)( 61,143)( 62,142)( 63,141)( 64,140)
( 65,139)( 66,138)( 67,137)( 68,203)( 69,202)( 70,268)( 71,267)( 72,266)
( 73,265)( 74,264)( 75,263)( 76,262)( 77,261)( 78,260)( 79,259)( 80,258)
( 81,257)( 82,256)( 83,255)( 84,254)( 85,253)( 86,252)( 87,251)( 88,250)
( 89,249)( 90,248)( 91,247)( 92,246)( 93,245)( 94,244)( 95,243)( 96,242)
( 97,241)( 98,240)( 99,239)(100,238)(101,237)(102,236)(103,235)(104,234)
(105,233)(106,232)(107,231)(108,230)(109,229)(110,228)(111,227)(112,226)
(113,225)(114,224)(115,223)(116,222)(117,221)(118,220)(119,219)(120,218)
(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)
(129,209)(130,208)(131,207)(132,206)(133,205)(134,204);;
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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(268)!(  2, 67)(  3, 66)(  4, 65)(  5, 64)(  6, 63)(  7, 62)(  8, 61)
(  9, 60)( 10, 59)( 11, 58)( 12, 57)( 13, 56)( 14, 55)( 15, 54)( 16, 53)
( 17, 52)( 18, 51)( 19, 50)( 20, 49)( 21, 48)( 22, 47)( 23, 46)( 24, 45)
( 25, 44)( 26, 43)( 27, 42)( 28, 41)( 29, 40)( 30, 39)( 31, 38)( 32, 37)
( 33, 36)( 34, 35)( 69,134)( 70,133)( 71,132)( 72,131)( 73,130)( 74,129)
( 75,128)( 76,127)( 77,126)( 78,125)( 79,124)( 80,123)( 81,122)( 82,121)
( 83,120)( 84,119)( 85,118)( 86,117)( 87,116)( 88,115)( 89,114)( 90,113)
( 91,112)( 92,111)( 93,110)( 94,109)( 95,108)( 96,107)( 97,106)( 98,105)
( 99,104)(100,103)(101,102)(135,202)(136,268)(137,267)(138,266)(139,265)
(140,264)(141,263)(142,262)(143,261)(144,260)(145,259)(146,258)(147,257)
(148,256)(149,255)(150,254)(151,253)(152,252)(153,251)(154,250)(155,249)
(156,248)(157,247)(158,246)(159,245)(160,244)(161,243)(162,242)(163,241)
(164,240)(165,239)(166,238)(167,237)(168,236)(169,235)(170,234)(171,233)
(172,232)(173,231)(174,230)(175,229)(176,228)(177,227)(178,226)(179,225)
(180,224)(181,223)(182,222)(183,221)(184,220)(185,219)(186,218)(187,217)
(188,216)(189,215)(190,214)(191,213)(192,212)(193,211)(194,210)(195,209)
(196,208)(197,207)(198,206)(199,205)(200,204)(201,203);
s1 := Sym(268)!(  1,136)(  2,135)(  3,201)(  4,200)(  5,199)(  6,198)(  7,197)
(  8,196)(  9,195)( 10,194)( 11,193)( 12,192)( 13,191)( 14,190)( 15,189)
( 16,188)( 17,187)( 18,186)( 19,185)( 20,184)( 21,183)( 22,182)( 23,181)
( 24,180)( 25,179)( 26,178)( 27,177)( 28,176)( 29,175)( 30,174)( 31,173)
( 32,172)( 33,171)( 34,170)( 35,169)( 36,168)( 37,167)( 38,166)( 39,165)
( 40,164)( 41,163)( 42,162)( 43,161)( 44,160)( 45,159)( 46,158)( 47,157)
( 48,156)( 49,155)( 50,154)( 51,153)( 52,152)( 53,151)( 54,150)( 55,149)
( 56,148)( 57,147)( 58,146)( 59,145)( 60,144)( 61,143)( 62,142)( 63,141)
( 64,140)( 65,139)( 66,138)( 67,137)( 68,203)( 69,202)( 70,268)( 71,267)
( 72,266)( 73,265)( 74,264)( 75,263)( 76,262)( 77,261)( 78,260)( 79,259)
( 80,258)( 81,257)( 82,256)( 83,255)( 84,254)( 85,253)( 86,252)( 87,251)
( 88,250)( 89,249)( 90,248)( 91,247)( 92,246)( 93,245)( 94,244)( 95,243)
( 96,242)( 97,241)( 98,240)( 99,239)(100,238)(101,237)(102,236)(103,235)
(104,234)(105,233)(106,232)(107,231)(108,230)(109,229)(110,228)(111,227)
(112,226)(113,225)(114,224)(115,223)(116,222)(117,221)(118,220)(119,219)
(120,218)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)
(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204);
poly := sub<Sym(268)|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*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope