Polytope of Type {12,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*1728l
if this polytope has a name.
Group : SmallGroup(1728,46587)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 12, 108, 36, 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 : {6,6,4}*864k
   4-fold quotients : {3,6,4}*432b
   9-fold quotients : {12,2,4}*192
   18-fold quotients : {12,2,2}*96, {6,2,4}*96
   27-fold quotients : {4,2,4}*64
   36-fold quotients : {3,2,4}*48, {6,2,2}*48
   54-fold quotients : {2,2,4}*32, {4,2,2}*32
   72-fold quotients : {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)
( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)
( 61, 85)( 62, 87)( 63, 86)( 64,100)( 65,102)( 66,101)( 67,106)( 68,108)
( 69,107)( 70,103)( 71,105)( 72,104)( 73, 91)( 74, 93)( 75, 92)( 76, 97)
( 77, 99)( 78, 98)( 79, 94)( 80, 96)( 81, 95);;
s1 := (  1, 67)(  2, 69)(  3, 68)(  4, 64)(  5, 66)(  6, 65)(  7, 70)(  8, 72)
(  9, 71)( 10, 58)( 11, 60)( 12, 59)( 13, 55)( 14, 57)( 15, 56)( 16, 61)
( 17, 63)( 18, 62)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)( 24, 74)
( 25, 79)( 26, 81)( 27, 80)( 28, 94)( 29, 96)( 30, 95)( 31, 91)( 32, 93)
( 33, 92)( 34, 97)( 35, 99)( 36, 98)( 37, 85)( 38, 87)( 39, 86)( 40, 82)
( 41, 84)( 42, 83)( 43, 88)( 44, 90)( 45, 89)( 46,103)( 47,105)( 48,104)
( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107);;
s2 := (  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);;
s3 := (  2,  6)(  3,  8)(  5,  9)( 11, 15)( 12, 17)( 14, 18)( 20, 24)( 21, 26)
( 23, 27)( 29, 33)( 30, 35)( 32, 36)( 38, 42)( 39, 44)( 41, 45)( 47, 51)
( 48, 53)( 50, 54)( 56, 60)( 57, 62)( 59, 63)( 65, 69)( 66, 71)( 68, 72)
( 74, 78)( 75, 80)( 77, 81)( 83, 87)( 84, 89)( 86, 90)( 92, 96)( 93, 98)
( 95, 99)(101,105)(102,107)(104,108);;
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(108)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)
( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)
( 43, 49)( 44, 51)( 45, 50)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)
( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64,100)( 65,102)( 66,101)( 67,106)
( 68,108)( 69,107)( 70,103)( 71,105)( 72,104)( 73, 91)( 74, 93)( 75, 92)
( 76, 97)( 77, 99)( 78, 98)( 79, 94)( 80, 96)( 81, 95);
s1 := Sym(108)!(  1, 67)(  2, 69)(  3, 68)(  4, 64)(  5, 66)(  6, 65)(  7, 70)
(  8, 72)(  9, 71)( 10, 58)( 11, 60)( 12, 59)( 13, 55)( 14, 57)( 15, 56)
( 16, 61)( 17, 63)( 18, 62)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)
( 24, 74)( 25, 79)( 26, 81)( 27, 80)( 28, 94)( 29, 96)( 30, 95)( 31, 91)
( 32, 93)( 33, 92)( 34, 97)( 35, 99)( 36, 98)( 37, 85)( 38, 87)( 39, 86)
( 40, 82)( 41, 84)( 42, 83)( 43, 88)( 44, 90)( 45, 89)( 46,103)( 47,105)
( 48,104)( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107);
s2 := 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);
s3 := Sym(108)!(  2,  6)(  3,  8)(  5,  9)( 11, 15)( 12, 17)( 14, 18)( 20, 24)
( 21, 26)( 23, 27)( 29, 33)( 30, 35)( 32, 36)( 38, 42)( 39, 44)( 41, 45)
( 47, 51)( 48, 53)( 50, 54)( 56, 60)( 57, 62)( 59, 63)( 65, 69)( 66, 71)
( 68, 72)( 74, 78)( 75, 80)( 77, 81)( 83, 87)( 84, 89)( 86, 90)( 92, 96)
( 93, 98)( 95, 99)(101,105)(102,107)(104,108);
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*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 >;

References : None.
to this polytope