Polytope of Type {2,2,6,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,9,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 6
Schlafli Type : {2,2,6,9,4}
Number of vertices, edges, etc : 2, 2, 6, 27, 18, 4
Order of s0s1s2s3s4s5 : 18
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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) :
3-fold quotients : {2,2,2,9,4}*576, {2,2,6,3,4}*576
9-fold quotients : {2,2,2,3,4}*192
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)
( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 53, 65)( 54, 66)( 55, 67)( 56, 68)
( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)( 63, 75)( 64, 76)
( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)
( 97,109)( 98,110)( 99,111)(100,112);;
s3 := ( 5, 17)( 6, 19)( 7, 18)( 8, 20)( 9, 25)( 10, 27)( 11, 26)( 12, 28)
( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 30, 31)( 33, 37)( 34, 39)( 35, 38)
( 36, 40)( 41, 93)( 42, 95)( 43, 94)( 44, 96)( 45, 89)( 46, 91)( 47, 90)
( 48, 92)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53, 81)( 54, 83)( 55, 82)
( 56, 84)( 57, 77)( 58, 79)( 59, 78)( 60, 80)( 61, 85)( 62, 87)( 63, 86)
( 64, 88)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)
( 72,104)( 73,109)( 74,111)( 75,110)( 76,112);;
s4 := ( 5, 41)( 6, 42)( 7, 44)( 8, 43)( 9, 49)( 10, 50)( 11, 52)( 12, 51)
( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 65)( 18, 66)( 19, 68)( 20, 67)
( 21, 73)( 22, 74)( 23, 76)( 24, 75)( 25, 69)( 26, 70)( 27, 72)( 28, 71)
( 29, 53)( 30, 54)( 31, 56)( 32, 55)( 33, 61)( 34, 62)( 35, 64)( 36, 63)
( 37, 57)( 38, 58)( 39, 60)( 40, 59)( 77, 81)( 78, 82)( 79, 84)( 80, 83)
( 87, 88)( 89,105)( 90,106)( 91,108)( 92,107)( 93,101)( 94,102)( 95,104)
( 96,103)( 97,109)( 98,110)( 99,112)(100,111);;
s5 := ( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)
( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)( 34, 35)
( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 52)( 50, 51)
( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)( 66, 67)
( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 84)( 82, 83)
( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)( 97,100)( 98, 99)
(101,104)(102,103)(105,108)(106,107)(109,112)(110,111);;
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,
s4*s5*s4*s5*s4*s5*s4*s5, s5*s4*s3*s5*s4*s5*s4*s3*s4,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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(112)!(1,2);
s1 := Sym(112)!(3,4);
s2 := Sym(112)!( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)( 23, 35)
( 24, 36)( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 53, 65)( 54, 66)( 55, 67)
( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)( 63, 75)
( 64, 76)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)( 95,107)
( 96,108)( 97,109)( 98,110)( 99,111)(100,112);
s3 := Sym(112)!( 5, 17)( 6, 19)( 7, 18)( 8, 20)( 9, 25)( 10, 27)( 11, 26)
( 12, 28)( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 30, 31)( 33, 37)( 34, 39)
( 35, 38)( 36, 40)( 41, 93)( 42, 95)( 43, 94)( 44, 96)( 45, 89)( 46, 91)
( 47, 90)( 48, 92)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53, 81)( 54, 83)
( 55, 82)( 56, 84)( 57, 77)( 58, 79)( 59, 78)( 60, 80)( 61, 85)( 62, 87)
( 63, 86)( 64, 88)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)
( 71,102)( 72,104)( 73,109)( 74,111)( 75,110)( 76,112);
s4 := Sym(112)!( 5, 41)( 6, 42)( 7, 44)( 8, 43)( 9, 49)( 10, 50)( 11, 52)
( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 65)( 18, 66)( 19, 68)
( 20, 67)( 21, 73)( 22, 74)( 23, 76)( 24, 75)( 25, 69)( 26, 70)( 27, 72)
( 28, 71)( 29, 53)( 30, 54)( 31, 56)( 32, 55)( 33, 61)( 34, 62)( 35, 64)
( 36, 63)( 37, 57)( 38, 58)( 39, 60)( 40, 59)( 77, 81)( 78, 82)( 79, 84)
( 80, 83)( 87, 88)( 89,105)( 90,106)( 91,108)( 92,107)( 93,101)( 94,102)
( 95,104)( 96,103)( 97,109)( 98,110)( 99,112)(100,111);
s5 := Sym(112)!( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)
( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)
( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 52)
( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)
( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 84)
( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)( 97,100)
( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111);
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, s4*s5*s4*s5*s4*s5*s4*s5,
s5*s4*s3*s5*s4*s5*s4*s3*s4, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope