Polytope of Type {9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,12}*432
if this polytope has a name.
Group : SmallGroup(432,522)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 18, 108, 24
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,12,2} of size 864
   {9,12,4} of size 1728
Vertex Figure Of :
   {2,9,12} of size 864
   {4,9,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,4}*144, {3,12}*144
   4-fold quotients : {9,6}*108
   6-fold quotients : {9,4}*72
   9-fold quotients : {3,4}*48
   12-fold quotients : {9,2}*36, {3,6}*36
   18-fold quotients : {3,4}*24
   36-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,24}*864, {18,12}*864b
   3-fold covers : {27,12}*1296, {9,36}*1296, {9,12}*1296c
   4-fold covers : {9,24}*1728, {36,12}*1728f, {36,12}*1728g, {18,24}*1728b, {18,24}*1728d, {18,12}*1728d, {9,12}*1728
Permutation Representation (GAP) :

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

References : None.
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