Polytope of Type {6,2,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,27}*648
if this polytope has a name.
Group : SmallGroup(648,298)
Rank : 4
Schlafli Type : {6,2,27}
Number of vertices, edges, etc : 6, 6, 27, 27
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,2,27,2} of size 1296
Vertex Figure Of :
   {2,6,2,27} of size 1296
   {3,6,2,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,27}*324
   3-fold quotients : {2,2,27}*216, {6,2,9}*216
   6-fold quotients : {3,2,9}*108
   9-fold quotients : {2,2,9}*72, {6,2,3}*72
   18-fold quotients : {3,2,3}*36
   27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,2,27}*1296, {6,2,54}*1296
   3-fold covers : {18,2,27}*1944, {6,6,27}*1944a, {6,2,81}*1944, {6,6,27}*1944b
Permutation Representation (GAP) :

s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)
(28,29)(30,31)(32,33);;
s3 := ( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)
(27,28)(29,30)(31,32);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(33)!(3,4)(5,6);
s1 := Sym(33)!(1,5)(2,3)(4,6);
s2 := Sym(33)!( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)
(26,27)(28,29)(30,31)(32,33);
s3 := Sym(33)!( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)
(25,26)(27,28)(29,30)(31,32);
poly := sub<Sym(33)|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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope