Polytope of Type {66,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {66,6}*1056
if this polytope has a name.
Group : SmallGroup(1056,1015)
Rank : 3
Schlafli Type : {66,6}
Number of vertices, edges, etc : 88, 264, 8
Order of s₀s₁s₂ : 44
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {33,6}*528
   11-fold quotients : {6,6}*96
   12-fold quotients : {22,2}*88
   22-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {11,2}*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)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 37)( 10, 38)( 11, 40)
( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)
( 20, 31)( 21, 25)( 22, 26)( 23, 28)( 24, 27)( 45, 89)( 46, 90)( 47, 92)
( 48, 91)( 49,129)( 50,130)( 51,132)( 52,131)( 53,125)( 54,126)( 55,128)
( 56,127)( 57,121)( 58,122)( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)
( 64,119)( 65,113)( 66,114)( 67,116)( 68,115)( 69,109)( 70,110)( 71,112)
( 72,111)( 73,105)( 74,106)( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)
( 80,103)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)
( 88, 95)(135,136)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)
(143,172)(144,171)(145,165)(146,166)(147,168)(148,167)(149,161)(150,162)
(151,164)(152,163)(153,157)(154,158)(155,160)(156,159)(177,221)(178,222)
(179,224)(180,223)(181,261)(182,262)(183,264)(184,263)(185,257)(186,258)
(187,260)(188,259)(189,253)(190,254)(191,256)(192,255)(193,249)(194,250)
(195,252)(196,251)(197,245)(198,246)(199,248)(200,247)(201,241)(202,242)
(203,244)(204,243)(205,237)(206,238)(207,240)(208,239)(209,233)(210,234)
(211,236)(212,235)(213,229)(214,230)(215,232)(216,231)(217,225)(218,226)
(219,228)(220,227);;
s1 := (  1,181)(  2,184)(  3,183)(  4,182)(  5,177)(  6,180)(  7,179)(  8,178)
(  9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)( 16,214)
( 17,209)( 18,212)( 19,211)( 20,210)( 21,205)( 22,208)( 23,207)( 24,206)
( 25,201)( 26,204)( 27,203)( 28,202)( 29,197)( 30,200)( 31,199)( 32,198)
( 33,193)( 34,196)( 35,195)( 36,194)( 37,189)( 38,192)( 39,191)( 40,190)
( 41,185)( 42,188)( 43,187)( 44,186)( 45,137)( 46,140)( 47,139)( 48,138)
( 49,133)( 50,136)( 51,135)( 52,134)( 53,173)( 54,176)( 55,175)( 56,174)
( 57,169)( 58,172)( 59,171)( 60,170)( 61,165)( 62,168)( 63,167)( 64,166)
( 65,161)( 66,164)( 67,163)( 68,162)( 69,157)( 70,160)( 71,159)( 72,158)
( 73,153)( 74,156)( 75,155)( 76,154)( 77,149)( 78,152)( 79,151)( 80,150)
( 81,145)( 82,148)( 83,147)( 84,146)( 85,141)( 86,144)( 87,143)( 88,142)
( 89,225)( 90,228)( 91,227)( 92,226)( 93,221)( 94,224)( 95,223)( 96,222)
( 97,261)( 98,264)( 99,263)(100,262)(101,257)(102,260)(103,259)(104,258)
(105,253)(106,256)(107,255)(108,254)(109,249)(110,252)(111,251)(112,250)
(113,245)(114,248)(115,247)(116,246)(117,241)(118,244)(119,243)(120,242)
(121,237)(122,240)(123,239)(124,238)(125,233)(126,236)(127,235)(128,234)
(129,229)(130,232)(131,231)(132,230);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)
( 33, 34)( 37, 38)( 41, 42)( 45, 90)( 46, 89)( 47, 91)( 48, 92)( 49, 94)
( 50, 93)( 51, 95)( 52, 96)( 53, 98)( 54, 97)( 55, 99)( 56,100)( 57,102)
( 58,101)( 59,103)( 60,104)( 61,106)( 62,105)( 63,107)( 64,108)( 65,110)
( 66,109)( 67,111)( 68,112)( 69,114)( 70,113)( 71,115)( 72,116)( 73,118)
( 74,117)( 75,119)( 76,120)( 77,122)( 78,121)( 79,123)( 80,124)( 81,126)
( 82,125)( 83,127)( 84,128)( 85,130)( 86,129)( 87,131)( 88,132)(133,134)
(137,138)(141,142)(145,146)(149,150)(153,154)(157,158)(161,162)(165,166)
(169,170)(173,174)(177,222)(178,221)(179,223)(180,224)(181,226)(182,225)
(183,227)(184,228)(185,230)(186,229)(187,231)(188,232)(189,234)(190,233)
(191,235)(192,236)(193,238)(194,237)(195,239)(196,240)(197,242)(198,241)
(199,243)(200,244)(201,246)(202,245)(203,247)(204,248)(205,250)(206,249)
(207,251)(208,252)(209,254)(210,253)(211,255)(212,256)(213,258)(214,257)
(215,259)(216,260)(217,262)(218,261)(219,263)(220,264);;
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*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*s2*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope