Polytope of Type {10,54}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope