Polytope of Type {2,2,4,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,4,6,9}*1728
if this polytope has a name.
Group : SmallGroup(1728,30872)
Rank : 6
Schlafli Type : {2,2,4,6,9}
Number of vertices, edges, etc : 2, 2, 4, 12, 27, 9
Order of s0s1s2s3s4s5 : 36
Order of s0s1s2s3s4s5s4s3s2s1 : 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 : {2,2,2,6,9}*864
3-fold quotients : {2,2,4,2,9}*576, {2,2,4,6,3}*576
6-fold quotients : {2,2,2,2,9}*288, {2,2,2,6,3}*288
9-fold quotients : {2,2,4,2,3}*192
18-fold quotients : {2,2,2,2,3}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)
( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)
( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)
( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)
( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)
( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)
( 53,107)( 54,108)( 55,109)( 56,110)( 57,111)( 58,112);;
s3 := ( 8, 11)( 9, 12)( 10, 13)( 17, 20)( 18, 21)( 19, 22)( 26, 29)( 27, 30)
( 28, 31)( 35, 38)( 36, 39)( 37, 40)( 44, 47)( 45, 48)( 46, 49)( 53, 56)
( 54, 57)( 55, 58)( 59, 86)( 60, 87)( 61, 88)( 62, 92)( 63, 93)( 64, 94)
( 65, 89)( 66, 90)( 67, 91)( 68, 95)( 69, 96)( 70, 97)( 71,101)( 72,102)
( 73,103)( 74, 98)( 75, 99)( 76,100)( 77,104)( 78,105)( 79,106)( 80,110)
( 81,111)( 82,112)( 83,107)( 84,108)( 85,109);;
s4 := ( 5, 8)( 6, 10)( 7, 9)( 12, 13)( 14, 27)( 15, 26)( 16, 28)( 17, 24)
( 18, 23)( 19, 25)( 20, 30)( 21, 29)( 22, 31)( 32, 35)( 33, 37)( 34, 36)
( 39, 40)( 41, 54)( 42, 53)( 43, 55)( 44, 51)( 45, 50)( 46, 52)( 47, 57)
( 48, 56)( 49, 58)( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 81)( 69, 80)
( 70, 82)( 71, 78)( 72, 77)( 73, 79)( 74, 84)( 75, 83)( 76, 85)( 86, 89)
( 87, 91)( 88, 90)( 93, 94)( 95,108)( 96,107)( 97,109)( 98,105)( 99,104)
(100,106)(101,111)(102,110)(103,112);;
s5 := ( 5, 14)( 6, 16)( 7, 15)( 8, 20)( 9, 22)( 10, 21)( 11, 17)( 12, 19)
( 13, 18)( 23, 24)( 26, 30)( 27, 29)( 28, 31)( 32, 41)( 33, 43)( 34, 42)
( 35, 47)( 36, 49)( 37, 48)( 38, 44)( 39, 46)( 40, 45)( 50, 51)( 53, 57)
( 54, 56)( 55, 58)( 59, 68)( 60, 70)( 61, 69)( 62, 74)( 63, 76)( 64, 75)
( 65, 71)( 66, 73)( 67, 72)( 77, 78)( 80, 84)( 81, 83)( 82, 85)( 86, 95)
( 87, 97)( 88, 96)( 89,101)( 90,103)( 91,102)( 92, 98)( 93,100)( 94, 99)
(104,105)(107,111)(108,110)(109,112);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(112)!(1,2);
s1 := Sym(112)!(3,4);
s2 := Sym(112)!( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)
( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)
( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)
( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)
( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)
( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)
( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)( 57,111)( 58,112);
s3 := Sym(112)!( 8, 11)( 9, 12)( 10, 13)( 17, 20)( 18, 21)( 19, 22)( 26, 29)
( 27, 30)( 28, 31)( 35, 38)( 36, 39)( 37, 40)( 44, 47)( 45, 48)( 46, 49)
( 53, 56)( 54, 57)( 55, 58)( 59, 86)( 60, 87)( 61, 88)( 62, 92)( 63, 93)
( 64, 94)( 65, 89)( 66, 90)( 67, 91)( 68, 95)( 69, 96)( 70, 97)( 71,101)
( 72,102)( 73,103)( 74, 98)( 75, 99)( 76,100)( 77,104)( 78,105)( 79,106)
( 80,110)( 81,111)( 82,112)( 83,107)( 84,108)( 85,109);
s4 := Sym(112)!( 5, 8)( 6, 10)( 7, 9)( 12, 13)( 14, 27)( 15, 26)( 16, 28)
( 17, 24)( 18, 23)( 19, 25)( 20, 30)( 21, 29)( 22, 31)( 32, 35)( 33, 37)
( 34, 36)( 39, 40)( 41, 54)( 42, 53)( 43, 55)( 44, 51)( 45, 50)( 46, 52)
( 47, 57)( 48, 56)( 49, 58)( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 81)
( 69, 80)( 70, 82)( 71, 78)( 72, 77)( 73, 79)( 74, 84)( 75, 83)( 76, 85)
( 86, 89)( 87, 91)( 88, 90)( 93, 94)( 95,108)( 96,107)( 97,109)( 98,105)
( 99,104)(100,106)(101,111)(102,110)(103,112);
s5 := Sym(112)!( 5, 14)( 6, 16)( 7, 15)( 8, 20)( 9, 22)( 10, 21)( 11, 17)
( 12, 19)( 13, 18)( 23, 24)( 26, 30)( 27, 29)( 28, 31)( 32, 41)( 33, 43)
( 34, 42)( 35, 47)( 36, 49)( 37, 48)( 38, 44)( 39, 46)( 40, 45)( 50, 51)
( 53, 57)( 54, 56)( 55, 58)( 59, 68)( 60, 70)( 61, 69)( 62, 74)( 63, 76)
( 64, 75)( 65, 71)( 66, 73)( 67, 72)( 77, 78)( 80, 84)( 81, 83)( 82, 85)
( 86, 95)( 87, 97)( 88, 96)( 89,101)( 90,103)( 91,102)( 92, 98)( 93,100)
( 94, 99)(104,105)(107,111)(108,110)(109,112);
poly := sub<Sym(112)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5,
s3*s5*s3*s5, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;
to this polytope