Polytope of Type {2,9,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6,4}*864
if this polytope has a name.
Group : SmallGroup(864,2501)
Rank : 5
Schlafli Type : {2,9,6,4}
Number of vertices, edges, etc : 2, 9, 27, 12, 4
Order of s0s1s2s3s4 : 36
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,9,6,4,2} of size 1728
Vertex Figure Of :
{2,2,9,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,9,6,2}*432
3-fold quotients : {2,9,2,4}*288, {2,3,6,4}*288
6-fold quotients : {2,9,2,2}*144, {2,3,6,2}*144
9-fold quotients : {2,3,2,4}*96
18-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,9,6,8}*1728, {2,18,6,4}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 22)( 13, 21)( 14, 23)( 15, 28)
( 16, 27)( 17, 29)( 18, 25)( 19, 24)( 20, 26)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 39, 49)( 40, 48)( 41, 50)( 42, 55)( 43, 54)( 44, 56)( 45, 52)
( 46, 51)( 47, 53)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 76)( 67, 75)
( 68, 77)( 69, 82)( 70, 81)( 71, 83)( 72, 79)( 73, 78)( 74, 80)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 93,103)( 94,102)( 95,104)( 96,109)( 97,108)
( 98,110)( 99,106)(100,105)(101,107);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 12)( 7, 14)( 8, 13)( 9, 18)( 10, 20)
( 11, 19)( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 42)( 31, 44)( 32, 43)
( 33, 39)( 34, 41)( 35, 40)( 36, 45)( 37, 47)( 38, 46)( 48, 52)( 49, 51)
( 50, 53)( 54, 55)( 57, 69)( 58, 71)( 59, 70)( 60, 66)( 61, 68)( 62, 67)
( 63, 72)( 64, 74)( 65, 73)( 75, 79)( 76, 78)( 77, 80)( 81, 82)( 84, 96)
( 85, 98)( 86, 97)( 87, 93)( 88, 95)( 89, 94)( 90, 99)( 91,101)( 92,100)
(102,106)(103,105)(104,107)(108,109);;
s3 := ( 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)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)
( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)( 70,100)
( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)( 78,108)
( 79,109)( 80,110)( 81,105)( 82,106)( 83,107);;
s4 := ( 3, 57)( 4, 58)( 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);;
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 22)( 13, 21)( 14, 23)
( 15, 28)( 16, 27)( 17, 29)( 18, 25)( 19, 24)( 20, 26)( 31, 32)( 33, 36)
( 34, 38)( 35, 37)( 39, 49)( 40, 48)( 41, 50)( 42, 55)( 43, 54)( 44, 56)
( 45, 52)( 46, 51)( 47, 53)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 76)
( 67, 75)( 68, 77)( 69, 82)( 70, 81)( 71, 83)( 72, 79)( 73, 78)( 74, 80)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 93,103)( 94,102)( 95,104)( 96,109)
( 97,108)( 98,110)( 99,106)(100,105)(101,107);
s2 := Sym(110)!( 3, 15)( 4, 17)( 5, 16)( 6, 12)( 7, 14)( 8, 13)( 9, 18)
( 10, 20)( 11, 19)( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 42)( 31, 44)
( 32, 43)( 33, 39)( 34, 41)( 35, 40)( 36, 45)( 37, 47)( 38, 46)( 48, 52)
( 49, 51)( 50, 53)( 54, 55)( 57, 69)( 58, 71)( 59, 70)( 60, 66)( 61, 68)
( 62, 67)( 63, 72)( 64, 74)( 65, 73)( 75, 79)( 76, 78)( 77, 80)( 81, 82)
( 84, 96)( 85, 98)( 86, 97)( 87, 93)( 88, 95)( 89, 94)( 90, 99)( 91,101)
( 92,100)(102,106)(103,105)(104,107)(108,109);
s3 := 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)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)
( 62, 92)( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)
( 70,100)( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)
( 78,108)( 79,109)( 80,110)( 81,105)( 82,106)( 83,107);
s4 := Sym(110)!( 3, 57)( 4, 58)( 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);
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope