Polytope of Type {6,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*1944
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 243, 486, 162
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {6,4}*216
   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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      81 vertex figures:
         81 of {4}*8
   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
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1> of order 3.
      60 facets:
         51 of {6}*12
         9 of {2}*4
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 9.
      24 facets:
         9 of {2}*4
         15 of {6}*12
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
      18 facets:
         18 of {6}*12
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 27.
      10 facets:
         6 of {2}*4
         4 of {6}*12
      9 vertex figures:
         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, 13)(  3, 14)(  4, 10)(  5, 11)(  6, 12)(  7, 17)(  8, 18)(  9, 16)( 19, 23)( 20, 24)( 21, 22)( 28, 86)( 29, 87)( 30, 85)( 31, 84)( 32, 82)( 33, 83)( 34, 88)( 35, 89)( 36, 90)( 37,105)( 38,103)( 39,104)( 40,100)( 41,101)( 42,102)( 43,107)( 44,108)( 45,106)( 46, 94)( 47, 95)( 48, 96)( 49, 92)( 50, 93)( 51, 91)( 52, 99)( 53, 97)( 54, 98)( 55,181)( 56,182)( 57,183)( 58,188)( 59,189)( 60,187)( 61,186)( 62,184)( 63,185)( 64,173)( 65,174)( 66,172)( 67,180)( 68,178)( 69,179)( 70,175)( 71,176)( 72,177)( 73,165)( 74,163)( 75,164)( 76,169)( 77,170)( 78,171)( 79,167)( 80,168)( 81,166)(112,116)(113,117)(114,115)(118,128)(119,129)(120,127)(121,135)(122,133)(123,134)(124,130)(125,131)(126,132)(136,198)(137,196)(138,197)(139,193)(140,194)(141,195)(142,191)(143,192)(144,190)(145,214)(146,215)(147,216)(148,212)(149,213)(150,211)(151,210)(152,208)(153,209)(154,206)(155,207)(156,205)(157,204)(158,202)(159,203)(160,199)(161,200)(162,201)(217,232)(218,233)(219,234)(220,230)(221,231)(222,229)(223,228)(224,226)(225,227)(235,243)(236,241)(237,242);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s2*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, 13)(  3, 14)(  4, 10)(  5, 11)(  6, 12)(  7, 17)(  8, 18)(  9, 16)( 19, 23)( 20, 24)( 21, 22)( 28, 86)( 29, 87)( 30, 85)( 31, 84)( 32, 82)( 33, 83)( 34, 88)( 35, 89)( 36, 90)( 37,105)( 38,103)( 39,104)( 40,100)( 41,101)( 42,102)( 43,107)( 44,108)( 45,106)( 46, 94)( 47, 95)( 48, 96)( 49, 92)( 50, 93)( 51, 91)( 52, 99)( 53, 97)( 54, 98)( 55,181)( 56,182)( 57,183)( 58,188)( 59,189)( 60,187)( 61,186)( 62,184)( 63,185)( 64,173)( 65,174)( 66,172)( 67,180)( 68,178)( 69,179)( 70,175)( 71,176)( 72,177)( 73,165)( 74,163)( 75,164)( 76,169)( 77,170)( 78,171)( 79,167)( 80,168)( 81,166)(112,116)(113,117)(114,115)(118,128)(119,129)(120,127)(121,135)(122,133)(123,134)(124,130)(125,131)(126,132)(136,198)(137,196)(138,197)(139,193)(140,194)(141,195)(142,191)(143,192)(144,190)(145,214)(146,215)(147,216)(148,212)(149,213)(150,211)(151,210)(152,208)(153,209)(154,206)(155,207)(156,205)(157,204)(158,202)(159,203)(160,199)(161,200)(162,201)(217,232)(218,233)(219,234)(220,230)(221,231)(222,229)(223,228)(224,226)(225,227)(235,243)(236,241)(237,242);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s2*s1 >;

References : None.
to this polytope

Polytope of Type {6,4}

Twisty Puzzle