Polytope of Type {54,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,10}*1080
Also Known As : {54,10|2}. if this polytope has another name.
Group : SmallGroup(1080,57)
Rank : 3
Schlafli Type : {54,10}
Number of vertices, edges, etc : 54, 270, 10
Order of s₀s₁s₂ : 270
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) :
   3-fold quotients : {18,10}*360
   5-fold quotients : {54,2}*216
   9-fold quotients : {6,10}*120
   10-fold quotients : {27,2}*108
   15-fold quotients : {18,2}*72
   27-fold quotients : {2,10}*40
   30-fold quotients : {9,2}*36
   45-fold quotients : {6,2}*24
   54-fold quotients : {2,5}*20
   90-fold quotients : {3,2}*12
   135-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)( 15, 18)
( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)( 33, 36)
( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 46, 95)( 47, 94)( 48, 96)( 49, 92)
( 50, 91)( 51, 93)( 52, 99)( 53, 98)( 54, 97)( 55,104)( 56,103)( 57,105)
( 58,101)( 59,100)( 60,102)( 61,108)( 62,107)( 63,106)( 64,113)( 65,112)
( 66,114)( 67,110)( 68,109)( 69,111)( 70,117)( 71,116)( 72,115)( 73,122)
( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,126)( 80,125)( 81,124)
( 82,131)( 83,130)( 84,132)( 85,128)( 86,127)( 87,129)( 88,135)( 89,134)
( 90,133)(137,138)(139,143)(140,142)(141,144)(146,147)(148,152)(149,151)
(150,153)(155,156)(157,161)(158,160)(159,162)(164,165)(166,170)(167,169)
(168,171)(173,174)(175,179)(176,178)(177,180)(181,230)(182,229)(183,231)
(184,227)(185,226)(186,228)(187,234)(188,233)(189,232)(190,239)(191,238)
(192,240)(193,236)(194,235)(195,237)(196,243)(197,242)(198,241)(199,248)
(200,247)(201,249)(202,245)(203,244)(204,246)(205,252)(206,251)(207,250)
(208,257)(209,256)(210,258)(211,254)(212,253)(213,255)(214,261)(215,260)
(216,259)(217,266)(218,265)(219,267)(220,263)(221,262)(222,264)(223,270)
(224,269)(225,268);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 53)(  5, 52)(  6, 54)(  7, 50)(  8, 49)
(  9, 51)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)( 16, 86)
( 17, 85)( 18, 87)( 19, 73)( 20, 75)( 21, 74)( 22, 80)( 23, 79)( 24, 81)
( 25, 77)( 26, 76)( 27, 78)( 28, 64)( 29, 66)( 30, 65)( 31, 71)( 32, 70)
( 33, 72)( 34, 68)( 35, 67)( 36, 69)( 37, 55)( 38, 57)( 39, 56)( 40, 62)
( 41, 61)( 42, 63)( 43, 59)( 44, 58)( 45, 60)( 91, 95)( 92, 94)( 93, 96)
( 97, 99)(100,131)(101,130)(102,132)(103,128)(104,127)(105,129)(106,135)
(107,134)(108,133)(109,122)(110,121)(111,123)(112,119)(113,118)(114,120)
(115,126)(116,125)(117,124)(136,181)(137,183)(138,182)(139,188)(140,187)
(141,189)(142,185)(143,184)(144,186)(145,217)(146,219)(147,218)(148,224)
(149,223)(150,225)(151,221)(152,220)(153,222)(154,208)(155,210)(156,209)
(157,215)(158,214)(159,216)(160,212)(161,211)(162,213)(163,199)(164,201)
(165,200)(166,206)(167,205)(168,207)(169,203)(170,202)(171,204)(172,190)
(173,192)(174,191)(175,197)(176,196)(177,198)(178,194)(179,193)(180,195)
(226,230)(227,229)(228,231)(232,234)(235,266)(236,265)(237,267)(238,263)
(239,262)(240,264)(241,270)(242,269)(243,268)(244,257)(245,256)(246,258)
(247,254)(248,253)(249,255)(250,261)(251,260)(252,259);;
s2 := (  1,145)(  2,146)(  3,147)(  4,148)(  5,149)(  6,150)(  7,151)(  8,152)
(  9,153)( 10,136)( 11,137)( 12,138)( 13,139)( 14,140)( 15,141)( 16,142)
( 17,143)( 18,144)( 19,172)( 20,173)( 21,174)( 22,175)( 23,176)( 24,177)
( 25,178)( 26,179)( 27,180)( 28,163)( 29,164)( 30,165)( 31,166)( 32,167)
( 33,168)( 34,169)( 35,170)( 36,171)( 37,154)( 38,155)( 39,156)( 40,157)
( 41,158)( 42,159)( 43,160)( 44,161)( 45,162)( 46,190)( 47,191)( 48,192)
( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,181)( 56,182)
( 57,183)( 58,184)( 59,185)( 60,186)( 61,187)( 62,188)( 63,189)( 64,217)
( 65,218)( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,224)( 72,225)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)( 88,205)
( 89,206)( 90,207)( 91,235)( 92,236)( 93,237)( 94,238)( 95,239)( 96,240)
( 97,241)( 98,242)( 99,243)(100,226)(101,227)(102,228)(103,229)(104,230)
(105,231)(106,232)(107,233)(108,234)(109,262)(110,263)(111,264)(112,265)
(113,266)(114,267)(115,268)(116,269)(117,270)(118,253)(119,254)(120,255)
(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,244)(128,245)
(129,246)(130,247)(131,248)(132,249)(133,250)(134,251)(135,252);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(270)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)
( 15, 18)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 46, 95)( 47, 94)( 48, 96)
( 49, 92)( 50, 91)( 51, 93)( 52, 99)( 53, 98)( 54, 97)( 55,104)( 56,103)
( 57,105)( 58,101)( 59,100)( 60,102)( 61,108)( 62,107)( 63,106)( 64,113)
( 65,112)( 66,114)( 67,110)( 68,109)( 69,111)( 70,117)( 71,116)( 72,115)
( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,126)( 80,125)
( 81,124)( 82,131)( 83,130)( 84,132)( 85,128)( 86,127)( 87,129)( 88,135)
( 89,134)( 90,133)(137,138)(139,143)(140,142)(141,144)(146,147)(148,152)
(149,151)(150,153)(155,156)(157,161)(158,160)(159,162)(164,165)(166,170)
(167,169)(168,171)(173,174)(175,179)(176,178)(177,180)(181,230)(182,229)
(183,231)(184,227)(185,226)(186,228)(187,234)(188,233)(189,232)(190,239)
(191,238)(192,240)(193,236)(194,235)(195,237)(196,243)(197,242)(198,241)
(199,248)(200,247)(201,249)(202,245)(203,244)(204,246)(205,252)(206,251)
(207,250)(208,257)(209,256)(210,258)(211,254)(212,253)(213,255)(214,261)
(215,260)(216,259)(217,266)(218,265)(219,267)(220,263)(221,262)(222,264)
(223,270)(224,269)(225,268);
s1 := Sym(270)!(  1, 46)(  2, 48)(  3, 47)(  4, 53)(  5, 52)(  6, 54)(  7, 50)
(  8, 49)(  9, 51)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)
( 16, 86)( 17, 85)( 18, 87)( 19, 73)( 20, 75)( 21, 74)( 22, 80)( 23, 79)
( 24, 81)( 25, 77)( 26, 76)( 27, 78)( 28, 64)( 29, 66)( 30, 65)( 31, 71)
( 32, 70)( 33, 72)( 34, 68)( 35, 67)( 36, 69)( 37, 55)( 38, 57)( 39, 56)
( 40, 62)( 41, 61)( 42, 63)( 43, 59)( 44, 58)( 45, 60)( 91, 95)( 92, 94)
( 93, 96)( 97, 99)(100,131)(101,130)(102,132)(103,128)(104,127)(105,129)
(106,135)(107,134)(108,133)(109,122)(110,121)(111,123)(112,119)(113,118)
(114,120)(115,126)(116,125)(117,124)(136,181)(137,183)(138,182)(139,188)
(140,187)(141,189)(142,185)(143,184)(144,186)(145,217)(146,219)(147,218)
(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)(154,208)(155,210)
(156,209)(157,215)(158,214)(159,216)(160,212)(161,211)(162,213)(163,199)
(164,201)(165,200)(166,206)(167,205)(168,207)(169,203)(170,202)(171,204)
(172,190)(173,192)(174,191)(175,197)(176,196)(177,198)(178,194)(179,193)
(180,195)(226,230)(227,229)(228,231)(232,234)(235,266)(236,265)(237,267)
(238,263)(239,262)(240,264)(241,270)(242,269)(243,268)(244,257)(245,256)
(246,258)(247,254)(248,253)(249,255)(250,261)(251,260)(252,259);
s2 := Sym(270)!(  1,145)(  2,146)(  3,147)(  4,148)(  5,149)(  6,150)(  7,151)
(  8,152)(  9,153)( 10,136)( 11,137)( 12,138)( 13,139)( 14,140)( 15,141)
( 16,142)( 17,143)( 18,144)( 19,172)( 20,173)( 21,174)( 22,175)( 23,176)
( 24,177)( 25,178)( 26,179)( 27,180)( 28,163)( 29,164)( 30,165)( 31,166)
( 32,167)( 33,168)( 34,169)( 35,170)( 36,171)( 37,154)( 38,155)( 39,156)
( 40,157)( 41,158)( 42,159)( 43,160)( 44,161)( 45,162)( 46,190)( 47,191)
( 48,192)( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,181)
( 56,182)( 57,183)( 58,184)( 59,185)( 60,186)( 61,187)( 62,188)( 63,189)
( 64,217)( 65,218)( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,224)
( 72,225)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)
( 88,205)( 89,206)( 90,207)( 91,235)( 92,236)( 93,237)( 94,238)( 95,239)
( 96,240)( 97,241)( 98,242)( 99,243)(100,226)(101,227)(102,228)(103,229)
(104,230)(105,231)(106,232)(107,233)(108,234)(109,262)(110,263)(111,264)
(112,265)(113,266)(114,267)(115,268)(116,269)(117,270)(118,253)(119,254)
(120,255)(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,244)
(128,245)(129,246)(130,247)(131,248)(132,249)(133,250)(134,251)(135,252);
poly := sub<Sym(270)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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