Polytope of Type {6,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*972a
if this polytope has a name.
Group : SmallGroup(972,107)
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}*1944b
Permutation Representation (GAP) :

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