Polytope of Type {6,12}

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

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

References : None.
to this polytope