Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*432b
if this polytope has a name.
Group : SmallGroup(432,301)
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
   {6,12,4} of size 1728
   {6,12,4} of size 1728
Vertex Figure Of :
   {2,6,12} of size 864
   {3,6,12} of size 1296
   {4,6,12} of size 1728
   {4,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*216b
   3-fold quotients : {6,12}*144a
   4-fold quotients : {6,6}*108
   6-fold quotients : {6,6}*72a
   9-fold quotients : {2,12}*48, {6,4}*48a
   18-fold quotients : {2,6}*24, {6,2}*24
   27-fold quotients : {2,4}*16
   36-fold quotients : {2,3}*12, {3,2}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,24}*864b, {12,12}*864c
   3-fold covers : {18,12}*1296a, {6,36}*1296b, {6,12}*1296a, {6,12}*1296b, {18,12}*1296b, {6,36}*1296f, {18,12}*1296c, {6,36}*1296g, {6,12}*1296g
   4-fold covers : {6,48}*1728b, {12,12}*1728c, {12,24}*1728d, {24,12}*1728d, {12,24}*1728f, {24,12}*1728f, {12,12}*1728l, {6,12}*1728b
Permutation Representation (GAP) :

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