Polytope of Type {3,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6}*216b
if this polytope has a name.
Group : SmallGroup(216,162)
Rank : 4
Schlafli Type : {3,6,6}
Number of vertices, edges, etc : 3, 9, 18, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,6,2} of size 432
   {3,6,6,3} of size 648
   {3,6,6,4} of size 864
   {3,6,6,3} of size 864
   {3,6,6,4} of size 864
   {3,6,6,6} of size 1296
   {3,6,6,6} of size 1296
   {3,6,6,6} of size 1296
   {3,6,6,8} of size 1728
   {3,6,6,4} of size 1728
   {3,6,6,6} of size 1728
   {3,6,6,9} of size 1944
   {3,6,6,3} of size 1944
Vertex Figure Of :
   {2,3,6,6} of size 432
   {4,3,6,6} of size 864
   {6,3,6,6} of size 1296
   {4,3,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,6}*72, {3,6,2}*72
   6-fold quotients : {3,2,3}*36
   9-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,12}*432b, {6,6,6}*432g
   3-fold covers : {3,6,18}*648b, {9,6,6}*648b, {3,6,6}*648c, {3,6,6}*648d, {3,6,6}*648e
   4-fold covers : {3,6,24}*864b, {12,6,6}*864d, {6,6,12}*864e, {6,12,6}*864f, {3,6,6}*864, {3,12,6}*864b
   5-fold covers : {3,6,30}*1080b, {15,6,6}*1080b
   6-fold covers : {3,6,36}*1296b, {9,6,12}*1296b, {3,6,12}*1296c, {3,6,12}*1296d, {3,6,12}*1296e, {6,6,18}*1296e, {18,6,6}*1296e, {6,6,6}*1296c, {6,6,6}*1296o, {6,6,6}*1296p, {6,6,6}*1296s
   7-fold covers : {3,6,42}*1512b, {21,6,6}*1512b
   8-fold covers : {3,6,48}*1728b, {24,6,6}*1728d, {6,6,24}*1728e, {12,6,12}*1728f, {12,12,6}*1728f, {6,24,6}*1728g, {6,12,12}*1728g, {3,12,6}*1728, {3,24,6}*1728b, {3,6,12}*1728, {3,12,12}*1728b, {6,6,6}*1728f, {6,12,6}*1728h, {6,12,6}*1728l
   9-fold covers : {9,6,18}*1944b, {9,18,6}*1944, {3,6,18}*1944c, {3,6,18}*1944d, {9,6,6}*1944c, {9,6,6}*1944d, {3,6,18}*1944e, {9,6,6}*1944e, {3,6,6}*1944b, {3,6,6}*1944c, {3,6,6}*1944d, {3,6,54}*1944b, {27,6,6}*1944b, {3,6,6}*1944e, {3,6,6}*1944f, {3,6,6}*1944g, {9,6,6}*1944f, {9,6,6}*1944g, {9,6,6}*1944h, {3,6,6}*1944h, {3,18,6}*1944
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);;
s1 := ( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)
(20,23)(21,24);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)
(17,27)(18,26);;
s3 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26);;
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, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope