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Polytope of Type {4,6,2}

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

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