Polytope of Type {3,2,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,12}*576a
if this polytope has a name.
Group : SmallGroup(576,6139)
Rank : 5
Schlafli Type : {3,2,4,12}
Number of vertices, edges, etc : 3, 3, 4, 24, 12
Order of s₀s₁s₂s₃s₄ : 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,4,12,2} of size 1152
Vertex Figure Of :
   {2,3,2,4,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,2,12}*288, {3,2,4,6}*288a
   3-fold quotients : {3,2,4,4}*192
   4-fold quotients : {3,2,2,6}*144
   6-fold quotients : {3,2,2,4}*96, {3,2,4,2}*96
   8-fold quotients : {3,2,2,3}*72
   12-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,8,12}*1152a, {3,2,4,24}*1152a, {3,2,8,12}*1152b, {3,2,4,24}*1152b, {3,2,4,12}*1152a, {6,2,4,12}*1152a
   3-fold covers : {9,2,4,12}*1728a, {3,2,4,36}*1728a, {3,2,12,12}*1728a, {3,2,12,12}*1728b, {3,6,4,12}*1728
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 9)( 6,13)(11,18)(12,19)(14,22)(15,23);;
s3 := ( 4, 5)( 6,10)( 7,12)( 8,11)( 9,17)(13,16)(14,21)(15,20)(18,27)(19,26)
(22,25)(23,24);;
s4 := ( 4, 7)( 5,14)( 6,11)( 9,22)(10,20)(12,15)(13,18)(16,24)(17,26)(19,23);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(27)!(2,3);
s1 := Sym(27)!(1,2);
s2 := Sym(27)!( 5, 9)( 6,13)(11,18)(12,19)(14,22)(15,23);
s3 := Sym(27)!( 4, 5)( 6,10)( 7,12)( 8,11)( 9,17)(13,16)(14,21)(15,20)(18,27)
(19,26)(22,25)(23,24);
s4 := Sym(27)!( 4, 7)( 5,14)( 6,11)( 9,22)(10,20)(12,15)(13,18)(16,24)(17,26)
(19,23);
poly := sub<Sym(27)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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