Polytope of Type {18,3}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,3}*972a
if this polytope has a name.
Group : SmallGroup(972,100)
Rank : 3
Schlafli Type : {18,3}
Number of vertices, edges, etc : 162, 243, 27
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,3,2} of size 1944
Vertex Figure Of :
   {2,18,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3}*324, {18,3}*324
   9-fold quotients : {6,3}*108
   27-fold quotients : {6,3}*36
   81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*1944b
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.
      9 facets:
         9 of {18}*36
      54 vertex figures:
         54 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      9 facets:
         9 of {18}*36
      54 vertex figures:
         54 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 9.
      3 facets:
         3 of {18}*36
      18 vertex figures:
         18 of {3}*6

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