Polytope of Type {2,12,18,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,18,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30763)
Rank : 5
Schlafli Type : {2,12,18,2}
Number of vertices, edges, etc : 2, 12, 108, 18, 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 : {2,6,18,2}*864a
3-fold quotients : {2,4,18,2}*576a, {2,12,6,2}*576a
6-fold quotients : {2,2,18,2}*288, {2,6,6,2}*288a
9-fold quotients : {2,12,2,2}*192, {2,4,6,2}*192a
12-fold quotients : {2,2,9,2}*144
18-fold quotients : {2,2,6,2}*96, {2,6,2,2}*96
27-fold quotients : {2,4,2,2}*64
36-fold quotients : {2,2,3,2}*48, {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)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)( 51, 54)
( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)
( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)( 70,100)
( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)( 78,108)
( 79,109)( 80,110)( 81,105)( 82,106)( 83,107);;
s2 := ( 3, 60)( 4, 62)( 5, 61)( 6, 57)( 7, 59)( 8, 58)( 9, 63)( 10, 65)
( 11, 64)( 12, 79)( 13, 78)( 14, 80)( 15, 76)( 16, 75)( 17, 77)( 18, 82)
( 19, 81)( 20, 83)( 21, 70)( 22, 69)( 23, 71)( 24, 67)( 25, 66)( 26, 68)
( 27, 73)( 28, 72)( 29, 74)( 30, 87)( 31, 89)( 32, 88)( 33, 84)( 34, 86)
( 35, 85)( 36, 90)( 37, 92)( 38, 91)( 39,106)( 40,105)( 41,107)( 42,103)
( 43,102)( 44,104)( 45,109)( 46,108)( 47,110)( 48, 97)( 49, 96)( 50, 98)
( 51, 94)( 52, 93)( 53, 95)( 54,100)( 55, 99)( 56,101);;
s3 := ( 3, 12)( 4, 14)( 5, 13)( 6, 15)( 7, 17)( 8, 16)( 9, 18)( 10, 20)
( 11, 19)( 21, 22)( 24, 25)( 27, 28)( 30, 39)( 31, 41)( 32, 40)( 33, 42)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 49)( 51, 52)( 54, 55)
( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)( 64, 74)
( 65, 73)( 75, 76)( 78, 79)( 81, 82)( 84, 93)( 85, 95)( 86, 94)( 87, 96)
( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(102,103)(105,106)(108,109);;
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*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(112)!(1,2);
s1 := Sym(112)!( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)
( 51, 54)( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)
( 62, 92)( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)
( 70,100)( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)
( 78,108)( 79,109)( 80,110)( 81,105)( 82,106)( 83,107);
s2 := Sym(112)!( 3, 60)( 4, 62)( 5, 61)( 6, 57)( 7, 59)( 8, 58)( 9, 63)
( 10, 65)( 11, 64)( 12, 79)( 13, 78)( 14, 80)( 15, 76)( 16, 75)( 17, 77)
( 18, 82)( 19, 81)( 20, 83)( 21, 70)( 22, 69)( 23, 71)( 24, 67)( 25, 66)
( 26, 68)( 27, 73)( 28, 72)( 29, 74)( 30, 87)( 31, 89)( 32, 88)( 33, 84)
( 34, 86)( 35, 85)( 36, 90)( 37, 92)( 38, 91)( 39,106)( 40,105)( 41,107)
( 42,103)( 43,102)( 44,104)( 45,109)( 46,108)( 47,110)( 48, 97)( 49, 96)
( 50, 98)( 51, 94)( 52, 93)( 53, 95)( 54,100)( 55, 99)( 56,101);
s3 := Sym(112)!( 3, 12)( 4, 14)( 5, 13)( 6, 15)( 7, 17)( 8, 16)( 9, 18)
( 10, 20)( 11, 19)( 21, 22)( 24, 25)( 27, 28)( 30, 39)( 31, 41)( 32, 40)
( 33, 42)( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 49)( 51, 52)
( 54, 55)( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)
( 64, 74)( 65, 73)( 75, 76)( 78, 79)( 81, 82)( 84, 93)( 85, 95)( 86, 94)
( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(102,103)(105,106)
(108,109);
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*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope