Polytope of Type {12,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,4}*1728b
Also Known As : {{12,4|2},{4,4}₆}. if this polytope has another name.
Group : SmallGroup(1728,46587)
Rank : 4
Schlafli Type : {12,4,4}
Number of vertices, edges, etc : 12, 108, 36, 18
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
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 : {6,4,4}*864b
   3-fold quotients : {4,4,4}*576b
   6-fold quotients : {2,4,4}*288
   12-fold quotients : {2,4,4}*144
   18-fold quotients : {12,2,2}*96
   36-fold quotients : {6,2,2}*48
   54-fold quotients : {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 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)
( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)( 68,104)
( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)( 76, 94)
( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99);;
s1 := (  1, 64)(  2, 71)(  3, 69)(  4, 70)(  5, 68)(  6, 66)(  7, 67)(  8, 65)
(  9, 72)( 10, 55)( 11, 62)( 12, 60)( 13, 61)( 14, 59)( 15, 57)( 16, 58)
( 17, 56)( 18, 63)( 19, 73)( 20, 80)( 21, 78)( 22, 79)( 23, 77)( 24, 75)
( 25, 76)( 26, 74)( 27, 81)( 28, 91)( 29, 98)( 30, 96)( 31, 97)( 32, 95)
( 33, 93)( 34, 94)( 35, 92)( 36, 99)( 37, 82)( 38, 89)( 39, 87)( 40, 88)
( 41, 86)( 42, 84)( 43, 85)( 44, 83)( 45, 90)( 46,100)( 47,107)( 48,105)
( 49,106)( 50,104)( 51,102)( 52,103)( 53,101)( 54,108);;
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 := (  1,  5)(  2,  7)(  4,  8)( 10, 14)( 11, 16)( 13, 17)( 19, 23)( 20, 25)
( 22, 26)( 28, 32)( 29, 34)( 31, 35)( 37, 41)( 38, 43)( 40, 44)( 46, 50)
( 47, 52)( 49, 53)( 55, 59)( 56, 61)( 58, 62)( 64, 68)( 65, 70)( 67, 71)
( 73, 77)( 74, 79)( 76, 80)( 82, 86)( 83, 88)( 85, 89)( 91, 95)( 92, 97)
( 94, 98)(100,104)(101,106)(103,107);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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