Polytope of Type {18,12,2,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12,2,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30763)
Rank : 5
Schlafli Type : {18,12,2,2}
Number of vertices, edges, etc : 18, 108, 12, 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 : {18,6,2,2}*864a
3-fold quotients : {18,4,2,2}*576a, {6,12,2,2}*576a
6-fold quotients : {18,2,2,2}*288, {6,6,2,2}*288a
9-fold quotients : {2,12,2,2}*192, {6,4,2,2}*192a
12-fold quotients : {9,2,2,2}*144
18-fold quotients : {2,6,2,2}*96, {6,2,2,2}*96
27-fold quotients : {2,4,2,2}*64
36-fold quotients : {2,3,2,2}*48, {3,2,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)( 14, 22)
( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 32, 33)( 35, 36)( 37, 47)
( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)
( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 77)( 68, 76)
( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 83, 84)( 86, 87)( 89, 90)( 91,101)
( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108);;
s1 := ( 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, 91)( 56, 93)( 57, 92)( 58, 97)( 59, 99)( 60, 98)
( 61, 94)( 62, 96)( 63, 95)( 64, 82)( 65, 84)( 66, 83)( 67, 88)( 68, 90)
( 69, 89)( 70, 85)( 71, 87)( 72, 86)( 73,101)( 74,100)( 75,102)( 76,107)
( 77,106)( 78,108)( 79,104)( 80,103)( 81,105);;
s2 := ( 1, 58)( 2, 59)( 3, 60)( 4, 55)( 5, 56)( 6, 57)( 7, 61)( 8, 62)
( 9, 63)( 10, 67)( 11, 68)( 12, 69)( 13, 64)( 14, 65)( 15, 66)( 16, 70)
( 17, 71)( 18, 72)( 19, 76)( 20, 77)( 21, 78)( 22, 73)( 23, 74)( 24, 75)
( 25, 79)( 26, 80)( 27, 81)( 28, 85)( 29, 86)( 30, 87)( 31, 82)( 32, 83)
( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37, 94)( 38, 95)( 39, 96)( 40, 91)
( 41, 92)( 42, 93)( 43, 97)( 44, 98)( 45, 99)( 46,103)( 47,104)( 48,105)
( 49,100)( 50,101)( 51,102)( 52,106)( 53,107)( 54,108);;
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, s0*s1*s2*s1*s0*s1*s2*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,
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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(112)!( 2, 3)( 5, 6)( 8, 9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)
( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 32, 33)( 35, 36)
( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)
( 45, 54)( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 77)
( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 83, 84)( 86, 87)( 89, 90)
( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)
( 99,108);
s1 := 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, 91)( 56, 93)( 57, 92)( 58, 97)( 59, 99)
( 60, 98)( 61, 94)( 62, 96)( 63, 95)( 64, 82)( 65, 84)( 66, 83)( 67, 88)
( 68, 90)( 69, 89)( 70, 85)( 71, 87)( 72, 86)( 73,101)( 74,100)( 75,102)
( 76,107)( 77,106)( 78,108)( 79,104)( 80,103)( 81,105);
s2 := Sym(112)!( 1, 58)( 2, 59)( 3, 60)( 4, 55)( 5, 56)( 6, 57)( 7, 61)
( 8, 62)( 9, 63)( 10, 67)( 11, 68)( 12, 69)( 13, 64)( 14, 65)( 15, 66)
( 16, 70)( 17, 71)( 18, 72)( 19, 76)( 20, 77)( 21, 78)( 22, 73)( 23, 74)
( 24, 75)( 25, 79)( 26, 80)( 27, 81)( 28, 85)( 29, 86)( 30, 87)( 31, 82)
( 32, 83)( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37, 94)( 38, 95)( 39, 96)
( 40, 91)( 41, 92)( 42, 93)( 43, 97)( 44, 98)( 45, 99)( 46,103)( 47,104)
( 48,105)( 49,100)( 50,101)( 51,102)( 52,106)( 53,107)( 54,108);
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,
s0*s1*s2*s1*s0*s1*s2*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,
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*s0*s1*s0*s1*s0*s1*s0*s1 >;
to this polytope