Polytope of Type {6,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3}*216b
if this polytope has a name.
Group : SmallGroup(216,162)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 6, 18, 9, 3
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,3,2} of size 432
   {6,6,3,4} of size 864
   {6,6,3,6} of size 1296
   {6,6,3,4} of size 1728
Vertex Figure Of :
   {2,6,6,3} of size 432
   {3,6,6,3} of size 648
   {4,6,6,3} of size 864
   {3,6,6,3} of size 864
   {4,6,6,3} of size 864
   {6,6,6,3} of size 1296
   {6,6,6,3} of size 1296
   {6,6,6,3} of size 1296
   {8,6,6,3} of size 1728
   {4,6,6,3} of size 1728
   {6,6,6,3} of size 1728
   {9,6,6,3} of size 1944
   {3,6,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,3}*72, {6,2,3}*72
   6-fold quotients : {3,2,3}*36
   9-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6,3}*432b, {6,6,6}*432c
   3-fold covers : {6,6,9}*648b, {18,6,3}*648b, {6,6,3}*648c, {6,6,3}*648d, {6,6,3}*648e
   4-fold covers : {24,6,3}*864b, {6,6,12}*864c, {12,6,6}*864e, {6,12,6}*864g, {6,6,3}*864, {6,12,3}*864b
   5-fold covers : {6,6,15}*1080b, {30,6,3}*1080b
   6-fold covers : {36,6,3}*1296b, {12,6,9}*1296b, {12,6,3}*1296c, {12,6,3}*1296d, {12,6,3}*1296e, {6,6,18}*1296c, {18,6,6}*1296c, {6,6,6}*1296f, {6,6,6}*1296k, {6,6,6}*1296n, {6,6,6}*1296q
   7-fold covers : {6,6,21}*1512b, {42,6,3}*1512b
   8-fold covers : {48,6,3}*1728b, {6,6,24}*1728c, {24,6,6}*1728e, {12,6,12}*1728e, {6,12,12}*1728c, {6,24,6}*1728f, {12,12,6}*1728g, {6,12,3}*1728, {6,24,3}*1728b, {12,6,3}*1728, {12,12,3}*1728b, {6,6,6}*1728a, {6,12,6}*1728f, {6,12,6}*1728j
   9-fold covers : {6,18,9}*1944, {18,6,9}*1944b, {6,6,9}*1944c, {6,6,9}*1944d, {18,6,3}*1944c, {18,6,3}*1944d, {6,6,9}*1944e, {18,6,3}*1944e, {6,6,3}*1944b, {6,6,3}*1944c, {6,6,3}*1944d, {6,6,27}*1944b, {54,6,3}*1944b, {6,6,3}*1944e, {6,6,3}*1944f, {6,6,3}*1944g, {6,6,9}*1944f, {6,6,9}*1944g, {6,6,9}*1944h, {6,6,3}*1944h, {6,18,3}*1944
Permutation Representation (GAP) :

s0 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27);;
s1 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(22,25)
(23,26)(24,27);;
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);;
s3 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)(22,26)
(23,25)(24,27);;
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, 
s0*s1*s2*s1*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(27)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27);
s1 := Sym(27)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)
(22,25)(23,26)(24,27);
s2 := Sym(27)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)
(20,24)(21,23)(26,27);
s3 := Sym(27)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)
(22,26)(23,25)(24,27);
poly := sub<Sym(27)|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, s0*s1*s2*s1*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 >;

References : None.
to this polytope