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