Polytope of Type {12,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1944a
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 54, 486, 81
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {4,18}*648
   9-fold quotients : {12,6}*216b
   27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 3.
      27 facets:
         27 of {12}*24
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 3.
      27 facets:
         27 of {12}*24
      36 vertex figures:
         9 of {18}*36
         27 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      27 facets:
         27 of {12}*24
      18 vertex figures:
         18 of {18}*36

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

Twisty Puzzle