Polytope of Type {3,2,27}

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

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

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