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Polytope of Type {12,18,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18,2}*864c
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {12,18,2}
Number of vertices, edges, etc : 12, 108, 18, 2
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,18,2,2} of size 1728
Vertex Figure Of :
   {2,12,18,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,18,2}*288c, {12,6,2}*288d
   6-fold quotients : {4,9,2}*144
   9-fold quotients : {4,6,2}*96b
   18-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,18,2}*1728a
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)
( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)
( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)
( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 21)(  6, 23)(  7, 22)(  8, 24)
(  9, 17)( 10, 19)( 11, 18)( 12, 20)( 26, 27)( 29, 33)( 30, 35)( 31, 34)
( 32, 36)( 37, 89)( 38, 91)( 39, 90)( 40, 92)( 41, 85)( 42, 87)( 43, 86)
( 44, 88)( 45, 93)( 46, 95)( 47, 94)( 48, 96)( 49, 77)( 50, 79)( 51, 78)
( 52, 80)( 53, 73)( 54, 75)( 55, 74)( 56, 76)( 57, 81)( 58, 83)( 59, 82)
( 60, 84)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)
( 68,100)( 69,105)( 70,107)( 71,106)( 72,108);;
s2 := (  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 81)(  6, 84)(  7, 83)(  8, 82)
(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 85)( 14, 88)( 15, 87)( 16, 86)
( 17, 93)( 18, 96)( 19, 95)( 20, 94)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)( 32,106)
( 33,101)( 34,104)( 35,103)( 36,102)( 38, 40)( 41, 45)( 42, 48)( 43, 47)
( 44, 46)( 50, 52)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 62, 64)( 65, 69)
( 66, 72)( 67, 71)( 68, 70);;
s3 := (109,110);;
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, s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(110)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)
( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)
( 56, 66)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)
( 90,104)( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106);
s1 := Sym(110)!(  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 21)(  6, 23)(  7, 22)
(  8, 24)(  9, 17)( 10, 19)( 11, 18)( 12, 20)( 26, 27)( 29, 33)( 30, 35)
( 31, 34)( 32, 36)( 37, 89)( 38, 91)( 39, 90)( 40, 92)( 41, 85)( 42, 87)
( 43, 86)( 44, 88)( 45, 93)( 46, 95)( 47, 94)( 48, 96)( 49, 77)( 50, 79)
( 51, 78)( 52, 80)( 53, 73)( 54, 75)( 55, 74)( 56, 76)( 57, 81)( 58, 83)
( 59, 82)( 60, 84)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)
( 67, 98)( 68,100)( 69,105)( 70,107)( 71,106)( 72,108);
s2 := Sym(110)!(  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 81)(  6, 84)(  7, 83)
(  8, 82)(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 85)( 14, 88)( 15, 87)
( 16, 86)( 17, 93)( 18, 96)( 19, 95)( 20, 94)( 21, 89)( 22, 92)( 23, 91)
( 24, 90)( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)
( 32,106)( 33,101)( 34,104)( 35,103)( 36,102)( 38, 40)( 41, 45)( 42, 48)
( 43, 47)( 44, 46)( 50, 52)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 62, 64)
( 65, 69)( 66, 72)( 67, 71)( 68, 70);
s3 := Sym(110)!(109,110);
poly := sub<Sym(110)|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, 
s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2 >; 
 

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