Polytope of Type {54,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,6}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 3
Schlafli Type : {54,6}
Number of vertices, edges, etc : 162, 486, 18
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {54,6}*648a, {54,6}*648b, {18,6}*648i
   6-fold quotients : {27,6}*324
   9-fold quotients : {54,2}*216, {18,6}*216a, {18,6}*216b, {6,6}*216d
   18-fold quotients : {27,2}*108, {9,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {9,2}*36, {3,6}*36, {6,3}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<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> of order 2.
      12 facets:
         6 of {27}*54
         6 of {54}*108
      81 vertex figures:
         81 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3.
      6 facets:
         6 of {54}*108
      108 vertex figures:
         27 of {6}*12
         81 of {2}*4
   P/N, where N=<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*s2*s1*s0*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {54}*108
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<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*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {54}*108
      54 vertex figures:
         54 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)( 32, 34)( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)( 82,163)( 83,165)( 84,164)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,189)( 98,188)( 99,187)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,180)(107,179)(108,178)(109,190)(110,192)(111,191)(112,197)(113,196)(114,198)(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)(122,208)(123,210)(124,216)(125,215)(126,214)(127,203)(128,202)(129,204)(130,200)(131,199)(132,201)(133,207)(134,206)(135,205)(136,217)(137,219)(138,218)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,243)(152,242)(153,241)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,234)(161,233)(162,232);;
s1 := (  1, 91)(  2, 93)(  3, 92)(  4, 98)(  5, 97)(  6, 99)(  7, 95)(  8, 94)(  9, 96)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)( 16, 86)( 17, 85)( 18, 87)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)( 24,102)( 25,108)( 26,107)( 27,106)( 28,145)( 29,147)( 30,146)( 31,152)( 32,151)( 33,153)( 34,149)( 35,148)( 36,150)( 37,136)( 38,138)( 39,137)( 40,143)( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,158)( 47,157)( 48,159)( 49,155)( 50,154)( 51,156)( 52,162)( 53,161)( 54,160)( 55,118)( 56,120)( 57,119)( 58,125)( 59,124)( 60,126)( 61,122)( 62,121)( 63,123)( 64,109)( 65,111)( 66,110)( 67,116)( 68,115)( 69,117)( 70,113)( 71,112)( 72,114)( 73,131)( 74,130)( 75,132)( 76,128)( 77,127)( 78,129)( 79,135)( 80,134)( 81,133)(163,172)(164,174)(165,173)(166,179)(167,178)(168,180)(169,176)(170,175)(171,177)(181,185)(182,184)(183,186)(187,189)(190,226)(191,228)(192,227)(193,233)(194,232)(195,234)(196,230)(197,229)(198,231)(199,217)(200,219)(201,218)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)(208,239)(209,238)(210,240)(211,236)(212,235)(213,237)(214,243)(215,242)(216,241);;
s2 := (  1, 28)(  2, 29)(  3, 30)(  4, 31)(  5, 32)(  6, 33)(  7, 34)(  8, 35)(  9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)( 16, 43)( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)( 24, 51)( 25, 52)( 26, 53)( 27, 54)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)(162,243);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*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*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(243)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)( 32, 34)( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)( 82,163)( 83,165)( 84,164)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,189)( 98,188)( 99,187)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,180)(107,179)(108,178)(109,190)(110,192)(111,191)(112,197)(113,196)(114,198)(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)(122,208)(123,210)(124,216)(125,215)(126,214)(127,203)(128,202)(129,204)(130,200)(131,199)(132,201)(133,207)(134,206)(135,205)(136,217)(137,219)(138,218)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,243)(152,242)(153,241)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,234)(161,233)(162,232);
s1 := Sym(243)!(  1, 91)(  2, 93)(  3, 92)(  4, 98)(  5, 97)(  6, 99)(  7, 95)(  8, 94)(  9, 96)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)( 16, 86)( 17, 85)( 18, 87)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)( 24,102)( 25,108)( 26,107)( 27,106)( 28,145)( 29,147)( 30,146)( 31,152)( 32,151)( 33,153)( 34,149)( 35,148)( 36,150)( 37,136)( 38,138)( 39,137)( 40,143)( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,158)( 47,157)( 48,159)( 49,155)( 50,154)( 51,156)( 52,162)( 53,161)( 54,160)( 55,118)( 56,120)( 57,119)( 58,125)( 59,124)( 60,126)( 61,122)( 62,121)( 63,123)( 64,109)( 65,111)( 66,110)( 67,116)( 68,115)( 69,117)( 70,113)( 71,112)( 72,114)( 73,131)( 74,130)( 75,132)( 76,128)( 77,127)( 78,129)( 79,135)( 80,134)( 81,133)(163,172)(164,174)(165,173)(166,179)(167,178)(168,180)(169,176)(170,175)(171,177)(181,185)(182,184)(183,186)(187,189)(190,226)(191,228)(192,227)(193,233)(194,232)(195,234)(196,230)(197,229)(198,231)(199,217)(200,219)(201,218)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)(208,239)(209,238)(210,240)(211,236)(212,235)(213,237)(214,243)(215,242)(216,241);
s2 := Sym(243)!(  1, 28)(  2, 29)(  3, 30)(  4, 31)(  5, 32)(  6, 33)(  7, 34)(  8, 35)(  9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)( 16, 43)( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)( 24, 51)( 25, 52)( 26, 53)( 27, 54)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)(162,243);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*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*s0*s1 >;
References : None.
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Twisty Puzzle