Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728z
if this polytope has a name.
Group : SmallGroup(1728,47870)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
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) :
   2-fold quotients : {12,12}*864m
   12-fold quotients : {4,4}*144
   18-fold quotients : {6,6}*96
   24-fold quotients : {4,4}*72
   36-fold quotients : {3,6}*48, {6,3}*48
   72-fold quotients : {3,3}*24
   216-fold quotients : {2,2}*8
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*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      40 facets:
         8 of {6}*12
         32 of {12}*24
      40 vertex figures:
         32 of {12}*24
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      20 facets:
         4 of {6}*12
         16 of {12}*24
      20 vertex figures:
         16 of {12}*24
         4 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 6.
      16 facets:
         8 of {6}*12
         8 of {12}*24
      16 vertex figures:
         8 of {12}*24
         8 of {6}*12

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

Twisty Puzzle