Polytope of Type {12,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,12630)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 216, 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
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*864a
   3-fold quotients : {4,4}*576
   4-fold quotients : {12,4}*432a
   6-fold quotients : {4,4}*288
   8-fold quotients : {12,4}*216
   12-fold quotients : {4,4}*144
   24-fold quotients : {4,4}*72
   27-fold quotients : {4,4}*64
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   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.
      38 facets:
         4 of {6}*12
         34 of {12}*24
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 2.
      36 facets:
         36 of {12}*24
      114 vertex figures:
         102 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      36 facets:
         36 of {12}*24
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      19 facets:
         2 of {6}*12
         17 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      18 facets:
         18 of {12}*24
      57 vertex figures:
         51 of {4}*8
         6 of {2}*4
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      19 facets:
         2 of {6}*12
         17 of {12}*24
      57 vertex figures:
         51 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
      12 facets:
         12 of {12}*24
      42 vertex figures:
         30 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      14 facets:
         4 of {6}*12
         10 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {12}*24
      42 vertex figures:
         30 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      14 facets:
         4 of {6}*12
         10 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 12.
      7 facets:
         2 of {6}*12
         5 of {12}*24
      21 vertex figures:
         15 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 12.
      7 facets:
         2 of {6}*12
         5 of {12}*24
      21 vertex figures:
         15 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 12.
      6 facets:
         6 of {12}*24
      18 vertex figures:
         18 of {4}*8

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

Twisty Puzzle