Polytope of Type {12,18,2,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18,2,2}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30872)
Rank : 5
Schlafli Type : {12,18,2,2}
Number of vertices, edges, etc : 12, 108, 18, 2, 2
Order of s0s1s2s3s4 : 36
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 : {6,18,2,2}*864b
3-fold quotients : {4,18,2,2}*576a, {12,6,2,2}*576c
4-fold quotients : {6,9,2,2}*432
6-fold quotients : {2,18,2,2}*288, {6,6,2,2}*288b
9-fold quotients : {4,6,2,2}*192a
12-fold quotients : {2,9,2,2}*144, {6,3,2,2}*144
18-fold quotients : {2,6,2,2}*96
27-fold quotients : {4,2,2,2}*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 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105);;
s1 := ( 1, 58)( 2, 60)( 3, 59)( 4, 55)( 5, 57)( 6, 56)( 7, 61)( 8, 63)
( 9, 62)( 10, 77)( 11, 76)( 12, 78)( 13, 74)( 14, 73)( 15, 75)( 16, 80)
( 17, 79)( 18, 81)( 19, 68)( 20, 67)( 21, 69)( 22, 65)( 23, 64)( 24, 66)
( 25, 71)( 26, 70)( 27, 72)( 28, 85)( 29, 87)( 30, 86)( 31, 82)( 32, 84)
( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37,104)( 38,103)( 39,105)( 40,101)
( 41,100)( 42,102)( 43,107)( 44,106)( 45,108)( 46, 95)( 47, 94)( 48, 96)
( 49, 92)( 50, 91)( 51, 93)( 52, 98)( 53, 97)( 54, 99);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 47)( 49, 53)
( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,101)(103,107)(104,106)(105,108);;
s3 := (109,110);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4, s2*s0*s1*s0*s1*s2*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*s0*s1,
s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(112)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105);
s1 := Sym(112)!( 1, 58)( 2, 60)( 3, 59)( 4, 55)( 5, 57)( 6, 56)( 7, 61)
( 8, 63)( 9, 62)( 10, 77)( 11, 76)( 12, 78)( 13, 74)( 14, 73)( 15, 75)
( 16, 80)( 17, 79)( 18, 81)( 19, 68)( 20, 67)( 21, 69)( 22, 65)( 23, 64)
( 24, 66)( 25, 71)( 26, 70)( 27, 72)( 28, 85)( 29, 87)( 30, 86)( 31, 82)
( 32, 84)( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37,104)( 38,103)( 39,105)
( 40,101)( 41,100)( 42,102)( 43,107)( 44,106)( 45,108)( 46, 95)( 47, 94)
( 48, 96)( 49, 92)( 50, 91)( 51, 93)( 52, 98)( 53, 97)( 54, 99);
s2 := Sym(112)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 47)
( 49, 53)( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 74)( 76, 80)( 77, 79)( 78, 81)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,101)(103,107)(104,106)(105,108);
s3 := Sym(112)!(109,110);
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*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s2*s0*s1*s0*s1*s2*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*s0*s1,
s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope