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Polytope of Type {2,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6}*192
if this polytope has a name.
Group : SmallGroup(192,1537)
Rank : 4
Schlafli Type : {2,4,6}
Number of vertices, edges, etc : 2, 8, 24, 12
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,6,2} of size 384
   {2,4,6,3} of size 768
   {2,4,6,3} of size 768
   {2,4,6,4} of size 768
   {2,4,6,4} of size 768
   {2,4,6,4} of size 768
   {2,4,6,6} of size 1152
   {2,4,6,6} of size 1152
   {2,4,6,6} of size 1728
   {2,4,6,10} of size 1920
Vertex Figure Of :
   {2,2,4,6} of size 384
   {3,2,4,6} of size 576
   {4,2,4,6} of size 768
   {5,2,4,6} of size 960
   {6,2,4,6} of size 1152
   {7,2,4,6} of size 1344
   {9,2,4,6} of size 1728
   {10,2,4,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   4-fold quotients : {2,4,3}*48, {2,2,6}*48
   8-fold quotients : {2,2,3}*24
   12-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,12}*384b, {4,4,6}*384d, {2,4,6}*384b, {2,4,12}*384c, {2,8,6}*384b, {2,8,6}*384c
   3-fold covers : {2,4,18}*576, {6,4,6}*576a, {2,12,6}*576a, {2,12,6}*576b
   4-fold covers : {2,4,12}*768d, {4,4,6}*768e, {4,4,12}*768e, {4,4,12}*768f, {2,8,6}*768d, {2,8,6}*768e, {4,4,6}*768f, {2,4,6}*768a, {2,8,12}*768e, {2,8,12}*768f, {2,4,24}*768c, {2,4,24}*768d, {4,8,6}*768c, {2,8,6}*768f, {2,8,12}*768g, {2,8,12}*768h, {8,4,6}*768c, {2,8,6}*768g, {4,8,6}*768d, {2,4,6}*768b, {2,4,24}*768e, {2,4,12}*768e, {2,4,24}*768f
   5-fold covers : {10,4,6}*960, {2,20,6}*960c, {2,4,30}*960
   6-fold covers : {2,4,36}*1152b, {4,4,18}*1152d, {2,4,18}*1152b, {2,4,36}*1152c, {2,8,18}*1152b, {2,8,18}*1152c, {6,4,12}*1152b, {2,12,12}*1152f, {2,12,12}*1152g, {12,4,6}*1152c, {2,12,6}*1152b, {2,12,12}*1152i, {4,12,6}*1152g, {6,4,6}*1152a, {6,4,12}*1152d, {2,24,6}*1152b, {2,24,6}*1152c, {2,24,6}*1152d, {6,8,6}*1152a, {2,24,6}*1152e, {6,8,6}*1152c, {4,12,6}*1152j, {2,12,6}*1152f, {2,12,12}*1152k
   7-fold covers : {14,4,6}*1344, {2,28,6}*1344, {2,4,42}*1344
   9-fold covers : {2,4,54}*1728, {18,4,6}*1728a, {2,36,6}*1728, {6,4,18}*1728b, {2,12,18}*1728a, {2,12,18}*1728b, {6,12,6}*1728a, {2,12,6}*1728a, {2,12,6}*1728b, {6,12,6}*1728e, {6,12,6}*1728f, {6,12,6}*1728g, {6,12,6}*1728h, {2,12,6}*1728c
   10-fold covers : {10,4,12}*1920b, {2,20,12}*1920b, {20,4,6}*1920b, {2,20,6}*1920a, {4,20,6}*1920c, {10,4,6}*1920, {10,4,12}*1920c, {2,40,6}*1920b, {10,8,6}*1920a, {2,40,6}*1920c, {10,8,6}*1920b, {2,20,12}*1920c, {2,4,60}*1920b, {4,4,30}*1920d, {2,4,30}*1920b, {2,4,60}*1920c, {2,8,30}*1920b, {2,8,30}*1920c
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 8)( 4, 6)( 5,12)( 7, 9)(10,14)(11,13)(15,18)(16,17);;
s2 := ( 6,10)( 8,13)( 9,15)(12,17);;
s3 := ( 3, 5)( 4, 7)( 6, 9)( 8,12)(10,16)(11,15)(13,18)(14,17);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(18)!(1,2);
s1 := Sym(18)!( 3, 8)( 4, 6)( 5,12)( 7, 9)(10,14)(11,13)(15,18)(16,17);
s2 := Sym(18)!( 6,10)( 8,13)( 9,15)(12,17);
s3 := Sym(18)!( 3, 5)( 4, 7)( 6, 9)( 8,12)(10,16)(11,15)(13,18)(14,17);
poly := sub<Sym(18)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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