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Polytope of Type {6,4,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,4,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46671)
Rank : 5
Schlafli Type : {6,4,4,6}
Number of vertices, edges, etc : 9, 18, 12, 12, 6
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Universal
   Non-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 : {6,4,2,6}*864
   3-fold quotients : {6,4,4,2}*576
   4-fold quotients : {6,4,2,3}*432
   6-fold quotients : {6,4,2,2}*288
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107);;
s1 := (  1,  4)(  3,  9)(  5,  8)( 10, 13)( 12, 18)( 14, 17)( 19, 22)( 21, 27)
( 23, 26)( 28, 31)( 30, 36)( 32, 35)( 37, 40)( 39, 45)( 41, 44)( 46, 49)
( 48, 54)( 50, 53)( 55, 58)( 57, 63)( 59, 62)( 64, 67)( 66, 72)( 68, 71)
( 73, 76)( 75, 81)( 77, 80)( 82, 85)( 84, 90)( 86, 89)( 91, 94)( 93, 99)
( 95, 98)(100,103)(102,108)(104,107);;
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)( 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, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 73)( 11, 74)( 12, 75)( 13, 76)( 14, 77)( 15, 78)( 16, 79)
( 17, 80)( 18, 81)( 19, 64)( 20, 65)( 21, 66)( 22, 67)( 23, 68)( 24, 69)
( 25, 70)( 26, 71)( 27, 72)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37,100)( 38,101)( 39,102)( 40,103)
( 41,104)( 42,105)( 43,106)( 44,107)( 45,108)( 46, 91)( 47, 92)( 48, 93)
( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)( 54, 99);;
s4 := ( 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,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(108)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107);
s1 := Sym(108)!(  1,  4)(  3,  9)(  5,  8)( 10, 13)( 12, 18)( 14, 17)( 19, 22)
( 21, 27)( 23, 26)( 28, 31)( 30, 36)( 32, 35)( 37, 40)( 39, 45)( 41, 44)
( 46, 49)( 48, 54)( 50, 53)( 55, 58)( 57, 63)( 59, 62)( 64, 67)( 66, 72)
( 68, 71)( 73, 76)( 75, 81)( 77, 80)( 82, 85)( 84, 90)( 86, 89)( 91, 94)
( 93, 99)( 95, 98)(100,103)(102,108)(104,107);
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)( 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, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)
(  8, 62)(  9, 63)( 10, 73)( 11, 74)( 12, 75)( 13, 76)( 14, 77)( 15, 78)
( 16, 79)( 17, 80)( 18, 81)( 19, 64)( 20, 65)( 21, 66)( 22, 67)( 23, 68)
( 24, 69)( 25, 70)( 26, 71)( 27, 72)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37,100)( 38,101)( 39,102)
( 40,103)( 41,104)( 42,105)( 43,106)( 44,107)( 45,108)( 46, 91)( 47, 92)
( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)( 54, 99);
s4 := 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,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 >; 
 
References : None.
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