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