Polytope of Type {12,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*1728n
if this polytope has a name.
Group : SmallGroup(1728,47215)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 36, 108, 36, 4
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 : {12,6,2}*864i
   3-fold quotients : {4,6,4}*576a
   4-fold quotients : {12,6,2}*432c
   6-fold quotients : {4,6,2}*288
   9-fold quotients : {12,2,4}*192
   12-fold quotients : {4,6,2}*144
   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,  8)(  3,  6)(  4,  7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)( 14, 23)
( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)( 37, 46)
( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)( 45, 54)
( 56, 62)( 57, 60)( 58, 61)( 64, 73)( 65, 80)( 66, 78)( 67, 79)( 68, 77)
( 69, 75)( 70, 76)( 71, 74)( 72, 81)( 83, 89)( 84, 87)( 85, 88)( 91,100)
( 92,107)( 93,105)( 94,106)( 95,104)( 96,102)( 97,103)( 98,101)( 99,108);;
s1 := (  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)(  8, 15)
(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)
( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)
( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107);;
s2 := (  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 14)( 11, 13)( 12, 15)( 16, 17)
( 19, 23)( 20, 22)( 21, 24)( 25, 26)( 28, 32)( 29, 31)( 30, 33)( 34, 35)
( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 50)( 47, 49)( 48, 51)( 52, 53)
( 55, 86)( 56, 85)( 57, 87)( 58, 83)( 59, 82)( 60, 84)( 61, 89)( 62, 88)
( 63, 90)( 64, 95)( 65, 94)( 66, 96)( 67, 92)( 68, 91)( 69, 93)( 70, 98)
( 71, 97)( 72, 99)( 73,104)( 74,103)( 75,105)( 76,101)( 77,100)( 78,102)
( 79,107)( 80,106)( 81,108);;
s3 := (  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)
( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)
( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*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,  8)(  3,  6)(  4,  7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)
( 14, 23)( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)
( 37, 46)( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)
( 45, 54)( 56, 62)( 57, 60)( 58, 61)( 64, 73)( 65, 80)( 66, 78)( 67, 79)
( 68, 77)( 69, 75)( 70, 76)( 71, 74)( 72, 81)( 83, 89)( 84, 87)( 85, 88)
( 91,100)( 92,107)( 93,105)( 94,106)( 95,104)( 96,102)( 97,103)( 98,101)
( 99,108);
s1 := Sym(108)!(  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)
(  8, 15)(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)
( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)
( 51, 53)( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)
( 62, 69)( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)
( 85, 99)( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)
(105,107);
s2 := Sym(108)!(  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 14)( 11, 13)( 12, 15)
( 16, 17)( 19, 23)( 20, 22)( 21, 24)( 25, 26)( 28, 32)( 29, 31)( 30, 33)
( 34, 35)( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 50)( 47, 49)( 48, 51)
( 52, 53)( 55, 86)( 56, 85)( 57, 87)( 58, 83)( 59, 82)( 60, 84)( 61, 89)
( 62, 88)( 63, 90)( 64, 95)( 65, 94)( 66, 96)( 67, 92)( 68, 91)( 69, 93)
( 70, 98)( 71, 97)( 72, 99)( 73,104)( 74,103)( 75,105)( 76,101)( 77,100)
( 78,102)( 79,107)( 80,106)( 81,108);
s3 := Sym(108)!(  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)
(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)
( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)
( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)
( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*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.
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