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Polytope of Type {3,2,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,8,6}*576
if this polytope has a name.
Group : SmallGroup(576,6606)
Rank : 5
Schlafli Type : {3,2,8,6}
Number of vertices, edges, etc : 3, 3, 8, 24, 6
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,8,6,2} of size 1152
   {3,2,8,6,3} of size 1728
Vertex Figure Of :
   {2,3,2,8,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,4,6}*288a
   3-fold quotients : {3,2,8,2}*192
   4-fold quotients : {3,2,2,6}*144
   6-fold quotients : {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, {3,2,16,6}*1152, {6,2,8,6}*1152
   3-fold covers : {3,2,8,18}*1728, {9,2,8,6}*1728, {3,2,24,6}*1728a, {3,6,8,6}*1728, {3,2,24,6}*1728c
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 8)( 9,12)(10,13)(11,14)(15,18)(16,19)(17,20)(21,24)(22,25);;
s3 := ( 4, 5)( 6,10)( 7, 9)( 8,11)(12,16)(13,15)(14,17)(18,22)(19,21)(20,23)
(24,27)(25,26);;
s4 := ( 4, 6)( 5, 9)( 8,12)(11,15)(14,18)(17,21)(20,24)(23,26);;
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*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(27)!(2,3);
s1 := Sym(27)!(1,2);
s2 := Sym(27)!( 5, 8)( 9,12)(10,13)(11,14)(15,18)(16,19)(17,20)(21,24)(22,25);
s3 := Sym(27)!( 4, 5)( 6,10)( 7, 9)( 8,11)(12,16)(13,15)(14,17)(18,22)(19,21)
(20,23)(24,27)(25,26);
s4 := Sym(27)!( 4, 6)( 5, 9)( 8,12)(11,15)(14,18)(17,21)(20,24)(23,26);
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*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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