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}*864b
if this polytope has a name.
Group : SmallGroup(864,2282)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 108, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {12,4,2} of size 1728
Vertex Figure Of :
   {2,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*432a
   3-fold quotients : {12,4}*288
   4-fold quotients : {6,4}*216
   6-fold quotients : {6,4}*144
   12-fold quotients : {6,4}*72
   27-fold quotients : {4,4}*32
   54-fold quotients : {2,4}*16, {4,2}*16
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,4}*1728b, {12,8}*1728a, {24,4}*1728a, {24,4}*1728c, {12,8}*1728d
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 2.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      18 facets:
         18 of {12}*24
      60 vertex figures:
         48 of {4}*8
         12 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      16 facets:
         10 of {12}*24
         6 of {4}*8
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1> of order 6.
      6 facets:
         6 of {12}*24
      24 vertex figures:
         12 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      8 facets:
         3 of {4}*8
         5 of {12}*24
      24 vertex figures:
         12 of {4}*8
         12 of {2}*4

Permutation Representation (GAP) : 
s0 := (  1, 55)(  2, 56)(  3, 57)(  4, 63)(  5, 61)(  6, 62)(  7, 59)(  8, 60)(  9, 58)( 10, 73)( 11, 74)( 12, 75)( 13, 81)( 14, 79)( 15, 80)( 16, 77)( 17, 78)( 18, 76)( 19, 64)( 20, 65)( 21, 66)( 22, 72)( 23, 70)( 24, 71)( 25, 68)( 26, 69)( 27, 67)( 28, 82)( 29, 83)( 30, 84)( 31, 90)( 32, 88)( 33, 89)( 34, 86)( 35, 87)( 36, 85)( 37,100)( 38,101)( 39,102)( 40,108)( 41,106)( 42,107)( 43,104)( 44,105)( 45,103)( 46, 91)( 47, 92)( 48, 93)( 49, 99)( 50, 97)( 51, 98)( 52, 95)( 53, 96)( 54, 94);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4,  6)(  7, 23)(  8, 22)(  9, 24)( 11, 12)( 16, 19)( 17, 21)( 18, 20)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 33)( 34, 50)( 35, 49)( 36, 51)( 38, 39)( 43, 46)( 44, 48)( 45, 47)( 52, 54)( 55, 94)( 56, 96)( 57, 95)( 58, 87)( 59, 86)( 60, 85)( 61,104)( 62,103)( 63,105)( 64, 91)( 65, 93)( 66, 92)( 67, 82)( 68, 84)( 69, 83)( 70,100)( 71,102)( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76, 89)( 77, 88)( 78, 90)( 79,108)( 80,107)( 81,106);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107);;
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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(108)!(  1, 55)(  2, 56)(  3, 57)(  4, 63)(  5, 61)(  6, 62)(  7, 59)(  8, 60)(  9, 58)( 10, 73)( 11, 74)( 12, 75)( 13, 81)( 14, 79)( 15, 80)( 16, 77)( 17, 78)( 18, 76)( 19, 64)( 20, 65)( 21, 66)( 22, 72)( 23, 70)( 24, 71)( 25, 68)( 26, 69)( 27, 67)( 28, 82)( 29, 83)( 30, 84)( 31, 90)( 32, 88)( 33, 89)( 34, 86)( 35, 87)( 36, 85)( 37,100)( 38,101)( 39,102)( 40,108)( 41,106)( 42,107)( 43,104)( 44,105)( 45,103)( 46, 91)( 47, 92)( 48, 93)( 49, 99)( 50, 97)( 51, 98)( 52, 95)( 53, 96)( 54, 94);
s1 := Sym(108)!(  1, 13)(  2, 15)(  3, 14)(  4,  6)(  7, 23)(  8, 22)(  9, 24)( 11, 12)( 16, 19)( 17, 21)( 18, 20)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 33)( 34, 50)( 35, 49)( 36, 51)( 38, 39)( 43, 46)( 44, 48)( 45, 47)( 52, 54)( 55, 94)( 56, 96)( 57, 95)( 58, 87)( 59, 86)( 60, 85)( 61,104)( 62,103)( 63,105)( 64, 91)( 65, 93)( 66, 92)( 67, 82)( 68, 84)( 69, 83)( 70,100)( 71,102)( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76, 89)( 77, 88)( 78, 90)( 79,108)( 80,107)( 81,106);
s2 := Sym(108)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107);
poly := sub<Sym(108)|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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1 >;
References : None.
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