Polytope of Type {27,2,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,2,3}*324
if this polytope has a name.
Group : SmallGroup(324,38)
Rank : 4
Schlafli Type : {27,2,3}
Number of vertices, edges, etc : 27, 27, 3, 3
Order of s₀s₁s₂s₃ : 27
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {27,2,3,2} of size 648
   {27,2,3,3} of size 1296
   {27,2,3,4} of size 1296
   {27,2,3,6} of size 1944
Vertex Figure Of :
   {2,27,2,3} of size 648
   {4,27,2,3} of size 1296
   {6,27,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,2,3}*108
   9-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {27,2,6}*648, {54,2,3}*648
   3-fold covers : {27,2,9}*972, {27,6,3}*972, {81,2,3}*972
   4-fold covers : {27,2,12}*1296, {108,2,3}*1296, {54,2,6}*1296
   5-fold covers : {135,2,3}*1620, {27,2,15}*1620
   6-fold covers : {27,2,18}*1944, {54,2,9}*1944, {27,6,6}*1944a, {54,6,3}*1944a, {81,2,6}*1944, {162,2,3}*1944, {27,6,6}*1944b, {54,6,3}*1944b
Permutation Representation (GAP) :

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

to this polytope