Polytope of Type {12,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,3}*432a
if this polytope has a name.
Group : SmallGroup(432,301)
Rank : 4
Schlafli Type : {12,6,3}
Number of vertices, edges, etc : 12, 36, 9, 3
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,6,3,2} of size 864
   {12,6,3,4} of size 1728
Vertex Figure Of :
   {2,12,6,3} of size 864
   {4,12,6,3} of size 1728
   {4,12,6,3} of size 1728
   {4,12,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,3}*216a
   3-fold quotients : {12,2,3}*144
   4-fold quotients : {3,6,3}*108
   6-fold quotients : {6,2,3}*72
   9-fold quotients : {4,2,3}*48
   12-fold quotients : {3,2,3}*36
   18-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6,3}*864a, {12,6,6}*864a
   3-fold covers : {12,6,9}*1296a, {36,6,3}*1296a, {12,6,3}*1296a, {12,6,3}*1296b, {12,6,3}*1296c
   4-fold covers : {48,6,3}*1728a, {12,6,12}*1728a, {12,12,6}*1728a, {24,6,6}*1728a, {12,12,3}*1728a
Permutation Representation (GAP) :

s0 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)
( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)
( 79, 97)( 80, 99)( 81, 98);;
s1 := (  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);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)
( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108);;
s3 := (  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);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*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,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)
( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)
( 78, 95)( 79, 97)( 80, 99)( 81, 98);
s1 := 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);
s2 := Sym(108)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)
( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)
( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)
( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)
(107,108);
s3 := 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);
poly := sub<Sym(108)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope