Polytope of Type {6,27}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,27}*972b
if this polytope has a name.
Group : SmallGroup(972,112)
Rank : 3
Schlafli Type : {6,27}
Number of vertices, edges, etc : 18, 243, 81
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,27,2} of size 1944
Vertex Figure Of :
   {2,6,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,9}*324a
   9-fold quotients : {6,9}*108, {6,3}*108
   27-fold quotients : {2,9}*36, {6,3}*36
   81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,54}*1944c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {6}*12
      6 vertex figures:
         6 of {27}*54

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