Polytope of Type {12,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,9}*1296d
if this polytope has a name.
Group : SmallGroup(1296,1790)
Rank : 3
Schlafli Type : {12,9}
Number of vertices, edges, etc : 72, 324, 54
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,3}*432
   4-fold quotients : {6,9}*324b
   9-fold quotients : {12,3}*144
   12-fold quotients : {6,3}*108
   27-fold quotients : {4,3}*48
   36-fold quotients : {6,3}*36
   54-fold quotients : {4,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)
( 23, 29)( 24, 30)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 83)( 42, 84)
( 43, 81)( 44, 82)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 99)( 50,100)
( 51, 97)( 52, 98)( 53,107)( 54,108)( 55,105)( 56,106)( 57,103)( 58,104)
( 59,101)( 60,102)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 65, 95)( 66, 96)
( 67, 93)( 68, 94)( 69, 91)( 70, 92)( 71, 89)( 72, 90);;
s1 := (  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)(  8, 47)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 57)( 14, 58)( 15, 60)( 16, 59)
( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)
( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 61)( 30, 62)( 31, 64)( 32, 63)
( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 75, 76)( 77, 81)( 78, 82)( 79, 84)
( 80, 83)( 85, 93)( 86, 94)( 87, 96)( 88, 95)( 91, 92)( 97,101)( 98,102)
( 99,104)(100,103)(107,108);;
s2 := (  2,  4)(  5,  9)(  6, 12)(  7, 11)(  8, 10)( 14, 16)( 17, 21)( 18, 24)
( 19, 23)( 20, 22)( 26, 28)( 29, 33)( 30, 36)( 31, 35)( 32, 34)( 37, 97)
( 38,100)( 39, 99)( 40, 98)( 41,105)( 42,108)( 43,107)( 44,106)( 45,101)
( 46,104)( 47,103)( 48,102)( 49, 73)( 50, 76)( 51, 75)( 52, 74)( 53, 81)
( 54, 84)( 55, 83)( 56, 82)( 57, 77)( 58, 80)( 59, 79)( 60, 78)( 61, 85)
( 62, 88)( 63, 87)( 64, 86)( 65, 93)( 66, 96)( 67, 95)( 68, 94)( 69, 89)
( 70, 92)( 71, 91)( 72, 90);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(108)!(  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)
( 22, 32)( 23, 29)( 24, 30)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 83)
( 42, 84)( 43, 81)( 44, 82)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 99)
( 50,100)( 51, 97)( 52, 98)( 53,107)( 54,108)( 55,105)( 56,106)( 57,103)
( 58,104)( 59,101)( 60,102)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 65, 95)
( 66, 96)( 67, 93)( 68, 94)( 69, 91)( 70, 92)( 71, 89)( 72, 90);
s1 := Sym(108)!(  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)
(  8, 47)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 57)( 14, 58)( 15, 60)
( 16, 59)( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 49)( 22, 50)( 23, 52)
( 24, 51)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 61)( 30, 62)( 31, 64)
( 32, 63)( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 75, 76)( 77, 81)( 78, 82)
( 79, 84)( 80, 83)( 85, 93)( 86, 94)( 87, 96)( 88, 95)( 91, 92)( 97,101)
( 98,102)( 99,104)(100,103)(107,108);
s2 := Sym(108)!(  2,  4)(  5,  9)(  6, 12)(  7, 11)(  8, 10)( 14, 16)( 17, 21)
( 18, 24)( 19, 23)( 20, 22)( 26, 28)( 29, 33)( 30, 36)( 31, 35)( 32, 34)
( 37, 97)( 38,100)( 39, 99)( 40, 98)( 41,105)( 42,108)( 43,107)( 44,106)
( 45,101)( 46,104)( 47,103)( 48,102)( 49, 73)( 50, 76)( 51, 75)( 52, 74)
( 53, 81)( 54, 84)( 55, 83)( 56, 82)( 57, 77)( 58, 80)( 59, 79)( 60, 78)
( 61, 85)( 62, 88)( 63, 87)( 64, 86)( 65, 93)( 66, 96)( 67, 95)( 68, 94)
( 69, 89)( 70, 92)( 71, 91)( 72, 90);
poly := sub<Sym(108)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope