Polytope of Type {2,6,4,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,4,18}*1728
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {2,6,4,18}
Number of vertices, edges, etc : 2, 6, 12, 36, 18
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 : {2,6,2,18}*864
3-fold quotients : {2,2,4,18}*576a, {2,6,4,6}*576
4-fold quotients : {2,3,2,18}*432, {2,6,2,9}*432
6-fold quotients : {2,2,2,18}*288, {2,6,2,6}*288
8-fold quotients : {2,3,2,9}*216
9-fold quotients : {2,2,4,6}*192a, {2,6,4,2}*192a
12-fold quotients : {2,2,2,9}*144, {2,3,2,6}*144, {2,6,2,3}*144
18-fold quotients : {2,2,2,6}*96, {2,6,2,2}*96
24-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,4,2}*64
36-fold quotients : {2,2,2,3}*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)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)( 71, 74)
( 78, 81)( 79, 82)( 80, 83)( 87, 90)( 88, 91)( 89, 92)( 96, 99)( 97,100)
( 98,101)(105,108)(106,109)(107,110);;
s2 := ( 3, 6)( 4, 7)( 5, 8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)( 22, 25)
( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)( 48, 51)
( 49, 52)( 50, 53)( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)
( 63, 90)( 64, 91)( 65, 92)( 66, 96)( 67, 97)( 68, 98)( 69, 93)( 70, 94)
( 71, 95)( 72, 99)( 73,100)( 74,101)( 75,105)( 76,106)( 77,107)( 78,102)
( 79,103)( 80,104)( 81,108)( 82,109)( 83,110);;
s3 := ( 3, 57)( 4, 59)( 5, 58)( 6, 60)( 7, 62)( 8, 61)( 9, 63)( 10, 65)
( 11, 64)( 12, 77)( 13, 76)( 14, 75)( 15, 80)( 16, 79)( 17, 78)( 18, 83)
( 19, 82)( 20, 81)( 21, 68)( 22, 67)( 23, 66)( 24, 71)( 25, 70)( 26, 69)
( 27, 74)( 28, 73)( 29, 72)( 30, 84)( 31, 86)( 32, 85)( 33, 87)( 34, 89)
( 35, 88)( 36, 90)( 37, 92)( 38, 91)( 39,104)( 40,103)( 41,102)( 42,107)
( 43,106)( 44,105)( 45,110)( 46,109)( 47,108)( 48, 95)( 49, 94)( 50, 93)
( 51, 98)( 52, 97)( 53, 96)( 54,101)( 55,100)( 56, 99);;
s4 := ( 3, 12)( 4, 14)( 5, 13)( 6, 15)( 7, 17)( 8, 16)( 9, 18)( 10, 20)
( 11, 19)( 21, 23)( 24, 26)( 27, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 42)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 50)( 51, 53)( 54, 56)
( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)( 64, 74)
( 65, 73)( 75, 77)( 78, 80)( 81, 83)( 84, 93)( 85, 95)( 86, 94)( 87, 96)
( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(102,104)(105,107)(108,110);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!( 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)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)
( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 87, 90)( 88, 91)( 89, 92)( 96, 99)
( 97,100)( 98,101)(105,108)(106,109)(107,110);
s2 := Sym(110)!( 3, 6)( 4, 7)( 5, 8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)
( 22, 25)( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)
( 48, 51)( 49, 52)( 50, 53)( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)
( 62, 86)( 63, 90)( 64, 91)( 65, 92)( 66, 96)( 67, 97)( 68, 98)( 69, 93)
( 70, 94)( 71, 95)( 72, 99)( 73,100)( 74,101)( 75,105)( 76,106)( 77,107)
( 78,102)( 79,103)( 80,104)( 81,108)( 82,109)( 83,110);
s3 := Sym(110)!( 3, 57)( 4, 59)( 5, 58)( 6, 60)( 7, 62)( 8, 61)( 9, 63)
( 10, 65)( 11, 64)( 12, 77)( 13, 76)( 14, 75)( 15, 80)( 16, 79)( 17, 78)
( 18, 83)( 19, 82)( 20, 81)( 21, 68)( 22, 67)( 23, 66)( 24, 71)( 25, 70)
( 26, 69)( 27, 74)( 28, 73)( 29, 72)( 30, 84)( 31, 86)( 32, 85)( 33, 87)
( 34, 89)( 35, 88)( 36, 90)( 37, 92)( 38, 91)( 39,104)( 40,103)( 41,102)
( 42,107)( 43,106)( 44,105)( 45,110)( 46,109)( 47,108)( 48, 95)( 49, 94)
( 50, 93)( 51, 98)( 52, 97)( 53, 96)( 54,101)( 55,100)( 56, 99);
s4 := Sym(110)!( 3, 12)( 4, 14)( 5, 13)( 6, 15)( 7, 17)( 8, 16)( 9, 18)
( 10, 20)( 11, 19)( 21, 23)( 24, 26)( 27, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 42)( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 50)( 51, 53)
( 54, 56)( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)
( 64, 74)( 65, 73)( 75, 77)( 78, 80)( 81, 83)( 84, 93)( 85, 95)( 86, 94)
( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(102,104)(105,107)
(108,110);
poly := sub<Sym(110)|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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope