Polytope of Type {6,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*972c
if this polytope has a name.
Group : SmallGroup(972,113)
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}*1944f
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope