Polytope of Type {6,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*972b
if this polytope has a name.
Group : SmallGroup(972,112)
Rank : 3
Schlafli Type : {6,54}
Number of vertices, edges, etc : 9, 243, 81
Order of s₀s₁s₂ : 27
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,54,2} of size 1944
Vertex Figure Of :
   {2,6,54} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,18}*324a
   9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,54}*1944d
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope