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Polytope of Type {2,12,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,6,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {2,12,6,2}
Number of vertices, edges, etc : 2, 36, 108, 18, 2
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,6,6,2}*864c
   3-fold quotients : {2,12,6,2}*576b
   4-fold quotients : {2,3,6,2}*432
   6-fold quotients : {2,6,6,2}*288c
   9-fold quotients : {2,12,2,2}*192
   12-fold quotients : {2,3,6,2}*144
   18-fold quotients : {2,6,2,2}*96
   27-fold quotients : {2,4,2,2}*64
   36-fold quotients : {2,3,2,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)( 63, 87)( 64, 88)
( 65, 89)( 66,102)( 67,103)( 68,104)( 69,108)( 70,109)( 71,110)( 72,105)
( 73,106)( 74,107)( 75, 93)( 76, 94)( 77, 95)( 78, 99)( 79,100)( 80,101)
( 81, 96)( 82, 97)( 83, 98);;
s2 := (  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)(  8, 76)(  9, 81)( 10, 82)
( 11, 83)( 12, 70)( 13, 71)( 14, 69)( 15, 68)( 16, 66)( 17, 67)( 18, 72)
( 19, 73)( 20, 74)( 21, 61)( 22, 62)( 23, 60)( 24, 59)( 25, 57)( 26, 58)
( 27, 63)( 28, 64)( 29, 65)( 30,106)( 31,107)( 32,105)( 33,104)( 34,102)
( 35,103)( 36,108)( 37,109)( 38,110)( 39, 97)( 40, 98)( 41, 96)( 42, 95)
( 43, 93)( 44, 94)( 45, 99)( 46,100)( 47,101)( 48, 88)( 49, 89)( 50, 87)
( 51, 86)( 52, 84)( 53, 85)( 54, 90)( 55, 91)( 56, 92);;
s3 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109);;
s4 := (111,112);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(112)!(1,2);
s1 := Sym(112)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)
( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)
( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)
( 47, 53)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)( 63, 87)
( 64, 88)( 65, 89)( 66,102)( 67,103)( 68,104)( 69,108)( 70,109)( 71,110)
( 72,105)( 73,106)( 74,107)( 75, 93)( 76, 94)( 77, 95)( 78, 99)( 79,100)
( 80,101)( 81, 96)( 82, 97)( 83, 98);
s2 := Sym(112)!(  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)(  8, 76)(  9, 81)
( 10, 82)( 11, 83)( 12, 70)( 13, 71)( 14, 69)( 15, 68)( 16, 66)( 17, 67)
( 18, 72)( 19, 73)( 20, 74)( 21, 61)( 22, 62)( 23, 60)( 24, 59)( 25, 57)
( 26, 58)( 27, 63)( 28, 64)( 29, 65)( 30,106)( 31,107)( 32,105)( 33,104)
( 34,102)( 35,103)( 36,108)( 37,109)( 38,110)( 39, 97)( 40, 98)( 41, 96)
( 42, 95)( 43, 93)( 44, 94)( 45, 99)( 46,100)( 47,101)( 48, 88)( 49, 89)
( 50, 87)( 51, 86)( 52, 84)( 53, 85)( 54, 90)( 55, 91)( 56, 92);
s3 := Sym(112)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109);
s4 := Sym(112)!(111,112);
poly := sub<Sym(112)|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*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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