Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1944b
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 81, 486, 162
Order of s0s1s2 : 4
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
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) :
   9-fold quotients : {6,12}*216b
   27-fold quotients : {6,4}*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*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      33 vertex figures:
         24 of {12}*24
         9 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 9.
      18 facets:
         18 of {6}*12
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 9.
      18 facets:
         18 of {6}*12
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8

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

Twisty Puzzle