Polytope of Type {4,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*1728l
if this polytope has a name.
Group : SmallGroup(1728,46587)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 12, 36, 108, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,6}*864k
   4-fold quotients : {4,6,3}*432b
   9-fold quotients : {4,2,12}*192
   18-fold quotients : {2,2,12}*96, {4,2,6}*96
   27-fold quotients : {4,2,4}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48
   54-fold quotients : {2,2,4}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24
   108-fold quotients : {2,2,2}*16
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> of order 3.
      12 facets:
         12 of 3-fold non-regular quotient of {4,6}*144
      4 vertex figures:
         4 of {6,12}*144b
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3.
      12 facets:
         12 of 3-fold non-regular quotient of {4,6}*144
      8 vertex figures:
         2 of {6,12}*144b
         6 of {2,12}*48

Permutation Representation (GAP) : 
s0 := (  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)( 48, 51)( 49, 52)( 56, 62)( 57, 60)( 58, 61)( 65, 71)( 66, 69)( 67, 70)( 74, 80)( 75, 78)( 76, 79)( 83, 89)( 84, 87)( 85, 88)( 92, 98)( 93, 96)( 94, 97)(101,107)(102,105)(103,106);;
s1 := (  4,  9)(  5,  7)(  6,  8)( 13, 18)( 14, 16)( 15, 17)( 22, 27)( 23, 25)( 24, 26)( 31, 36)( 32, 34)( 33, 35)( 40, 45)( 41, 43)( 42, 44)( 49, 54)( 50, 52)( 51, 53)( 58, 63)( 59, 61)( 60, 62)( 67, 72)( 68, 70)( 69, 71)( 76, 81)( 77, 79)( 78, 80)( 85, 90)( 86, 88)( 87, 89)( 94, 99)( 95, 97)( 96, 98)(103,108)(104,106)(105,107);;
s2 := (  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 26)( 17, 25)( 18, 27)( 28, 32)( 29, 31)( 30, 33)( 34, 35)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 53)( 44, 52)( 45, 54)( 55, 86)( 56, 85)( 57, 87)( 58, 83)( 59, 82)( 60, 84)( 61, 89)( 62, 88)( 63, 90)( 64,104)( 65,103)( 66,105)( 67,101)( 68,100)( 69,102)( 70,107)( 71,106)( 72,108)( 73, 95)( 74, 94)( 75, 96)( 76, 92)( 77, 91)( 78, 93)( 79, 98)( 80, 97)( 81, 99);;
s3 := (  1, 64)(  2, 66)(  3, 65)(  4, 70)(  5, 72)(  6, 71)(  7, 67)(  8, 69)(  9, 68)( 10, 55)( 11, 57)( 12, 56)( 13, 61)( 14, 63)( 15, 62)( 16, 58)( 17, 60)( 18, 59)( 19, 73)( 20, 75)( 21, 74)( 22, 79)( 23, 81)( 24, 80)( 25, 76)( 26, 78)( 27, 77)( 28, 91)( 29, 93)( 30, 92)( 31, 97)( 32, 99)( 33, 98)( 34, 94)( 35, 96)( 36, 95)( 37, 82)( 38, 84)( 39, 83)( 40, 88)( 41, 90)( 42, 89)( 43, 85)( 44, 87)( 45, 86)( 46,100)( 47,102)( 48,101)( 49,106)( 50,108)( 51,107)( 52,103)( 53,105)( 54,104);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(108)!(  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)( 48, 51)( 49, 52)( 56, 62)( 57, 60)( 58, 61)( 65, 71)( 66, 69)( 67, 70)( 74, 80)( 75, 78)( 76, 79)( 83, 89)( 84, 87)( 85, 88)( 92, 98)( 93, 96)( 94, 97)(101,107)(102,105)(103,106);
s1 := Sym(108)!(  4,  9)(  5,  7)(  6,  8)( 13, 18)( 14, 16)( 15, 17)( 22, 27)( 23, 25)( 24, 26)( 31, 36)( 32, 34)( 33, 35)( 40, 45)( 41, 43)( 42, 44)( 49, 54)( 50, 52)( 51, 53)( 58, 63)( 59, 61)( 60, 62)( 67, 72)( 68, 70)( 69, 71)( 76, 81)( 77, 79)( 78, 80)( 85, 90)( 86, 88)( 87, 89)( 94, 99)( 95, 97)( 96, 98)(103,108)(104,106)(105,107);
s2 := Sym(108)!(  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 26)( 17, 25)( 18, 27)( 28, 32)( 29, 31)( 30, 33)( 34, 35)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 53)( 44, 52)( 45, 54)( 55, 86)( 56, 85)( 57, 87)( 58, 83)( 59, 82)( 60, 84)( 61, 89)( 62, 88)( 63, 90)( 64,104)( 65,103)( 66,105)( 67,101)( 68,100)( 69,102)( 70,107)( 71,106)( 72,108)( 73, 95)( 74, 94)( 75, 96)( 76, 92)( 77, 91)( 78, 93)( 79, 98)( 80, 97)( 81, 99);
s3 := Sym(108)!(  1, 64)(  2, 66)(  3, 65)(  4, 70)(  5, 72)(  6, 71)(  7, 67)(  8, 69)(  9, 68)( 10, 55)( 11, 57)( 12, 56)( 13, 61)( 14, 63)( 15, 62)( 16, 58)( 17, 60)( 18, 59)( 19, 73)( 20, 75)( 21, 74)( 22, 79)( 23, 81)( 24, 80)( 25, 76)( 26, 78)( 27, 77)( 28, 91)( 29, 93)( 30, 92)( 31, 97)( 32, 99)( 33, 98)( 34, 94)( 35, 96)( 36, 95)( 37, 82)( 38, 84)( 39, 83)( 40, 88)( 41, 90)( 42, 89)( 43, 85)( 44, 87)( 45, 86)( 46,100)( 47,102)( 48,101)( 49,106)( 50,108)( 51,107)( 52,103)( 53,105)( 54,104);
poly := sub<Sym(108)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
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