Polytope of Type {6,88}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,88}*1056
Also Known As : {6,88|2}. if this polytope has another name.
Group : SmallGroup(1056,240)
Rank : 3
Schlafli Type : {6,88}
Number of vertices, edges, etc : 6, 264, 88
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 : {6,44}*528a
   3-fold quotients : {2,88}*352
   4-fold quotients : {6,22}*264
   6-fold quotients : {2,44}*176
   11-fold quotients : {6,8}*96
   12-fold quotients : {2,22}*88
   22-fold quotients : {6,4}*48a
   24-fold quotients : {2,11}*44
   33-fold quotients : {2,8}*32
   44-fold quotients : {6,2}*24
   66-fold quotients : {2,4}*16
   88-fold quotients : {3,2}*12
   132-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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