Polytope of Type {9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,12}*1296d
if this polytope has a name.
Group : SmallGroup(1296,1790)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 54, 324, 72
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 : {3,12}*432
   4-fold quotients : {9,6}*324b
   9-fold quotients : {3,12}*144
   12-fold quotients : {3,6}*108
   27-fold quotients : {3,4}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {3,4}*24
   108-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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