Polytope of Type {12,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46671)
Rank : 4
Schlafli Type : {12,4,6}
Number of vertices, edges, etc : 36, 72, 36, 6
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 : {6,4,6}*864b
   3-fold quotients : {12,4,2}*576
   4-fold quotients : {6,4,6}*432a
   6-fold quotients : {6,4,2}*288
   9-fold quotients : {4,4,6}*192
   12-fold quotients : {6,4,2}*144
   18-fold quotients : {2,4,6}*96a, {4,2,6}*96
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48
   54-fold quotients : {2,4,2}*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.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope