Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*432a
if this polytope has a name.
Group : SmallGroup(432,545)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 18, 18, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,6,2} of size 864
   {6,6,6,4} of size 1728
   {6,6,6,4} of size 1728
   {6,6,6,4} of size 1728
Vertex Figure Of :
   {2,6,6,6} of size 864
   {4,6,6,6} of size 1728
   {4,6,6,6} of size 1728
   {4,6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,6}*216a, {6,6,3}*216a
   3-fold quotients : {6,2,6}*144
   4-fold quotients : {3,6,3}*108
   6-fold quotients : {3,2,6}*72, {6,2,3}*72
   9-fold quotients : {2,2,6}*48, {6,2,2}*48
   12-fold quotients : {3,2,3}*36
   18-fold quotients : {2,2,3}*24, {3,2,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6,12}*864a, {12,6,6}*864a, {6,12,6}*864a
   3-fold covers : {6,6,18}*1296a, {18,6,6}*1296a, {6,6,6}*1296a, {6,6,6}*1296b, {6,6,6}*1296n, {6,6,6}*1296p
   4-fold covers : {12,6,12}*1728a, {6,12,12}*1728a, {12,12,6}*1728a, {6,6,24}*1728a, {24,6,6}*1728a, {6,24,6}*1728a, {6,12,6}*1728a, {6,12,6}*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, 58)(  2, 60)(  3, 59)(  4, 55)(  5, 57)(  6, 56)(  7, 61)(  8, 63)
(  9, 62)( 10, 67)( 11, 69)( 12, 68)( 13, 64)( 14, 66)( 15, 65)( 16, 70)
( 17, 72)( 18, 71)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)( 24, 74)
( 25, 79)( 26, 81)( 27, 80)( 28, 85)( 29, 87)( 30, 86)( 31, 82)( 32, 84)
( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37, 94)( 38, 96)( 39, 95)( 40, 91)
( 41, 93)( 42, 92)( 43, 97)( 44, 99)( 45, 98)( 46,103)( 47,105)( 48,104)
( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)(  8, 17)
(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)
( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)( 62, 71)
( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 95)
( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)(106,108);;
s3 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)( 16, 52)
( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)
( 25, 43)( 26, 45)( 27, 44)( 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);;
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*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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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, 58)(  2, 60)(  3, 59)(  4, 55)(  5, 57)(  6, 56)(  7, 61)
(  8, 63)(  9, 62)( 10, 67)( 11, 69)( 12, 68)( 13, 64)( 14, 66)( 15, 65)
( 16, 70)( 17, 72)( 18, 71)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)
( 24, 74)( 25, 79)( 26, 81)( 27, 80)( 28, 85)( 29, 87)( 30, 86)( 31, 82)
( 32, 84)( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37, 94)( 38, 96)( 39, 95)
( 40, 91)( 41, 93)( 42, 92)( 43, 97)( 44, 99)( 45, 98)( 46,103)( 47,105)
( 48,104)( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107);
s2 := Sym(108)!(  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)
(  8, 17)(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)
( 62, 71)( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 95)( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)
(106,108);
s3 := Sym(108)!(  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)
(  8, 36)(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)
( 24, 41)( 25, 43)( 26, 45)( 27, 44)( 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);
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*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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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