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}*1728ab
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 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   2-fold quotients : {12,12}*864o
   4-fold quotients : {6,12}*432i
   8-fold quotients : {6,12}*216c
   9-fold quotients : {4,12}*192b
   12-fold quotients : {6,4}*144
   18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
   24-fold quotients : {6,4}*72
   36-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
   72-fold quotients : {4,3}*24, {2,6}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {2,3}*12
   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.
      48 facets:
         24 of {6}*12
         24 of {12}*24
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> 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*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*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*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 3.
      48 facets:
         12 of {12}*24
         36 of {4}*8
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      24 facets:
         12 of {6}*12
         12 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s1> of order 6.
      12 facets:
         12 of {12}*24
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      32 facets:
         4 of {6}*12
         12 of {2}*4
         12 of {4}*8
         4 of {12}*24
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 6.
      24 facets:
         6 of {12}*24
         18 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 6.
      16 facets:
         8 of {6}*12
         8 of {12}*24
      12 vertex figures:
         12 of {12}*24

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

Twisty Puzzle