Polytope of Type {6,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,4}*1728b
Also Known As : {{6,4|2},{4,4|6}}. if this polytope has another name.
Group : SmallGroup(1728,46671)
Rank : 4
Schlafli Type : {6,4,4}
Number of vertices, edges, etc : 6, 108, 72, 36
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 : {2,4,4}*576
   6-fold quotients : {2,4,4}*288
   9-fold quotients : {6,4,4}*192
   12-fold quotients : {2,4,4}*144
   18-fold quotients : {6,2,4}*96, {6,4,2}*96a
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {3,2,4}*48, {6,2,2}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*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)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108);;
s1 := (  1, 10)(  2, 17)(  3, 15)(  4, 16)(  5, 14)(  6, 12)(  7, 13)(  8, 11)
(  9, 18)( 20, 26)( 21, 24)( 22, 25)( 28, 37)( 29, 44)( 30, 42)( 31, 43)
( 32, 41)( 33, 39)( 34, 40)( 35, 38)( 36, 45)( 47, 53)( 48, 51)( 49, 52)
( 55, 64)( 56, 71)( 57, 69)( 58, 70)( 59, 68)( 60, 66)( 61, 67)( 62, 65)
( 63, 72)( 74, 80)( 75, 78)( 76, 79)( 82, 91)( 83, 98)( 84, 96)( 85, 97)
( 86, 95)( 87, 93)( 88, 94)( 89, 92)( 90, 99)(101,107)(102,105)(103,106);;
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)( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)( 60, 89)
( 61, 86)( 62, 87)( 63, 85)( 64, 91)( 65, 92)( 66, 93)( 67, 99)( 68, 97)
( 69, 98)( 70, 95)( 71, 96)( 72, 94)( 73,100)( 74,101)( 75,102)( 76,108)
( 77,106)( 78,107)( 79,104)( 80,105)( 81,103);;
s3 := (  1, 59)(  2, 61)(  3, 57)(  4, 62)(  5, 55)(  6, 60)(  7, 56)(  8, 58)
(  9, 63)( 10, 68)( 11, 70)( 12, 66)( 13, 71)( 14, 64)( 15, 69)( 16, 65)
( 17, 67)( 18, 72)( 19, 77)( 20, 79)( 21, 75)( 22, 80)( 23, 73)( 24, 78)
( 25, 74)( 26, 76)( 27, 81)( 28, 86)( 29, 88)( 30, 84)( 31, 89)( 32, 82)
( 33, 87)( 34, 83)( 35, 85)( 36, 90)( 37, 95)( 38, 97)( 39, 93)( 40, 98)
( 41, 91)( 42, 96)( 43, 92)( 44, 94)( 45, 99)( 46,104)( 47,106)( 48,102)
( 49,107)( 50,100)( 51,105)( 52,101)( 53,103)( 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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
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)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108);
s1 := Sym(108)!(  1, 10)(  2, 17)(  3, 15)(  4, 16)(  5, 14)(  6, 12)(  7, 13)
(  8, 11)(  9, 18)( 20, 26)( 21, 24)( 22, 25)( 28, 37)( 29, 44)( 30, 42)
( 31, 43)( 32, 41)( 33, 39)( 34, 40)( 35, 38)( 36, 45)( 47, 53)( 48, 51)
( 49, 52)( 55, 64)( 56, 71)( 57, 69)( 58, 70)( 59, 68)( 60, 66)( 61, 67)
( 62, 65)( 63, 72)( 74, 80)( 75, 78)( 76, 79)( 82, 91)( 83, 98)( 84, 96)
( 85, 97)( 86, 95)( 87, 93)( 88, 94)( 89, 92)( 90, 99)(101,107)(102,105)
(103,106);
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)( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)
( 60, 89)( 61, 86)( 62, 87)( 63, 85)( 64, 91)( 65, 92)( 66, 93)( 67, 99)
( 68, 97)( 69, 98)( 70, 95)( 71, 96)( 72, 94)( 73,100)( 74,101)( 75,102)
( 76,108)( 77,106)( 78,107)( 79,104)( 80,105)( 81,103);
s3 := Sym(108)!(  1, 59)(  2, 61)(  3, 57)(  4, 62)(  5, 55)(  6, 60)(  7, 56)
(  8, 58)(  9, 63)( 10, 68)( 11, 70)( 12, 66)( 13, 71)( 14, 64)( 15, 69)
( 16, 65)( 17, 67)( 18, 72)( 19, 77)( 20, 79)( 21, 75)( 22, 80)( 23, 73)
( 24, 78)( 25, 74)( 26, 76)( 27, 81)( 28, 86)( 29, 88)( 30, 84)( 31, 89)
( 32, 82)( 33, 87)( 34, 83)( 35, 85)( 36, 90)( 37, 95)( 38, 97)( 39, 93)
( 40, 98)( 41, 91)( 42, 96)( 43, 92)( 44, 94)( 45, 99)( 46,104)( 47,106)
( 48,102)( 49,107)( 50,100)( 51,105)( 52,101)( 53,103)( 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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;

References : None.
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