Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*432c
if this polytope has a name.
Group : SmallGroup(432,324)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 18, 108, 36
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,12,2} of size 864
   {6,12,4} of size 1728
Vertex Figure Of :
   {2,6,12} of size 864
   {4,6,12} of size 1728
   {4,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*216c
   3-fold quotients : {6,12}*144c
   4-fold quotients : {3,6}*108
   6-fold quotients : {6,6}*72c
   9-fold quotients : {6,4}*48a
   12-fold quotients : {3,6}*36
   18-fold quotients : {6,2}*24
   27-fold quotients : {2,4}*16
   36-fold quotients : {3,2}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12}*864a, {6,24}*864c
   3-fold covers : {18,12}*1296e, {18,12}*1296f, {18,12}*1296g, {18,12}*1296h, {6,12}*1296d, {6,36}*1296h, {6,12}*1296i
   4-fold covers : {12,24}*1728a, {12,12}*1728a, {12,24}*1728b, {24,12}*1728c, {24,12}*1728e, {6,48}*1728c, {6,12}*1728e, {6,12}*1728f
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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