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Polytope of Type {2,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,4}*96
if this polytope has a name.
Group : SmallGroup(96,226)
Rank : 4
Schlafli Type : {2,3,4}
Number of vertices, edges, etc : 2, 6, 12, 8
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Locally Projective
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,3,4,2} of size 192
   {2,3,4,4} of size 384
   {2,3,4,6} of size 576
   {2,3,4,4} of size 768
   {2,3,4,8} of size 768
   {2,3,4,10} of size 960
   {2,3,4,12} of size 1152
   {2,3,4,3} of size 1152
   {2,3,4,14} of size 1344
   {2,3,4,4} of size 1440
   {2,3,4,18} of size 1728
   {2,3,4,20} of size 1920
Vertex Figure Of :
   {2,2,3,4} of size 192
   {3,2,3,4} of size 288
   {4,2,3,4} of size 384
   {5,2,3,4} of size 480
   {6,2,3,4} of size 576
   {7,2,3,4} of size 672
   {8,2,3,4} of size 768
   {9,2,3,4} of size 864
   {10,2,3,4} of size 960
   {11,2,3,4} of size 1056
   {12,2,3,4} of size 1152
   {13,2,3,4} of size 1248
   {14,2,3,4} of size 1344
   {15,2,3,4} of size 1440
   {17,2,3,4} of size 1632
   {18,2,3,4} of size 1728
   {19,2,3,4} of size 1824
   {20,2,3,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,3,4}*48
   4-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,3,8}*192, {2,6,4}*192
   3-fold covers : {2,9,4}*288, {6,3,4}*288, {2,3,12}*288
   4-fold covers : {2,3,8}*384, {2,12,4}*384b, {4,6,4}*384a, {2,6,4}*384b, {2,12,4}*384c, {2,6,8}*384b, {2,6,8}*384c, {4,3,4}*384
   5-fold covers : {2,15,4}*480
   6-fold covers : {2,9,8}*576, {2,18,4}*576, {2,3,24}*576, {6,3,8}*576, {6,6,4}*576a, {6,6,4}*576b, {2,6,12}*576a, {2,6,12}*576b
   7-fold covers : {2,21,4}*672
   8-fold covers : {2,3,8}*768, {2,6,8}*768a, {4,12,4}*768e, {2,12,4}*768d, {4,6,4}*768c, {4,12,4}*768g, {2,6,8}*768d, {2,6,8}*768e, {2,6,4}*768a, {2,12,8}*768e, {2,12,8}*768f, {2,24,4}*768c, {2,24,4}*768d, {2,6,8}*768f, {2,12,8}*768g, {2,12,8}*768h, {8,6,4}*768a, {2,6,8}*768g, {4,6,8}*768b, {4,6,8}*768c, {2,6,4}*768b, {2,24,4}*768e, {2,12,4}*768e, {2,24,4}*768f, {4,3,8}*768b, {8,3,4}*768b, {4,3,8}*768c, {8,3,4}*768c, {4,3,4}*768, {4,6,4}*768j, {4,6,4}*768l
   9-fold covers : {2,27,4}*864, {6,9,4}*864, {2,9,12}*864, {2,3,12}*864, {6,3,4}*864, {6,3,12}*864
   10-fold covers : {2,15,8}*960, {10,6,4}*960e, {2,6,20}*960c, {2,30,4}*960
   11-fold covers : {2,33,4}*1056
   12-fold covers : {2,9,8}*1152, {2,36,4}*1152b, {4,18,4}*1152a, {2,18,4}*1152b, {2,36,4}*1152c, {2,18,8}*1152b, {2,18,8}*1152c, {2,3,24}*1152, {6,3,8}*1152, {4,9,4}*1152, {6,12,4}*1152e, {6,12,4}*1152f, {2,12,12}*1152d, {2,12,12}*1152e, {12,6,4}*1152a, {2,6,12}*1152b, {2,12,12}*1152h, {4,6,12}*1152b, {4,6,12}*1152c, {6,6,4}*1152c, {6,6,4}*1152d, {6,12,4}*1152g, {6,12,4}*1152h, {2,6,24}*1152b, {2,6,24}*1152c, {2,6,24}*1152d, {6,6,8}*1152b, {6,6,8}*1152c, {2,6,24}*1152e, {6,6,8}*1152d, {6,6,8}*1152e, {2,6,12}*1152f, {12,6,4}*1152d, {2,12,12}*1152j, {2,3,12}*1152, {6,3,4}*1152b, {4,3,12}*1152b, {12,3,4}*1152b
   13-fold covers : {2,39,4}*1248
   14-fold covers : {2,21,8}*1344, {14,6,4}*1344, {2,6,28}*1344, {2,42,4}*1344
   15-fold covers : {2,45,4}*1440, {6,15,4}*1440b, {2,15,12}*1440
   17-fold covers : {2,51,4}*1632
   18-fold covers : {2,27,8}*1728, {2,54,4}*1728, {2,9,24}*1728, {2,3,24}*1728, {6,9,8}*1728, {6,3,8}*1728, {18,6,4}*1728, {2,6,36}*1728, {6,18,4}*1728a, {6,18,4}*1728b, {2,18,12}*1728a, {2,18,12}*1728b, {6,6,4}*1728a, {6,6,4}*1728b, {2,6,12}*1728a, {2,6,12}*1728b, {6,3,24}*1728, {6,6,4}*1728c, {6,6,12}*1728a, {6,6,12}*1728b, {6,6,12}*1728c, {6,6,12}*1728d, {2,6,12}*1728c
   19-fold covers : {2,57,4}*1824
   20-fold covers : {2,15,8}*1920a, {10,12,4}*1920b, {2,12,20}*1920b, {20,6,4}*1920a, {2,6,20}*1920a, {4,6,20}*1920b, {10,6,4}*1920b, {10,12,4}*1920c, {2,6,40}*1920b, {10,6,8}*1920a, {2,6,40}*1920c, {10,6,8}*1920b, {2,12,20}*1920c, {2,60,4}*1920b, {4,30,4}*1920a, {2,30,4}*1920b, {2,60,4}*1920c, {2,30,8}*1920b, {2,30,8}*1920c, {4,15,4}*1920c
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,6)(4,8);;
s2 := (5,6)(7,8);;
s3 := (5,7);;
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, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(8)!(1,2);
s1 := Sym(8)!(3,6)(4,8);
s2 := Sym(8)!(5,6)(7,8);
s3 := Sym(8)!(5,7);
poly := sub<Sym(8)|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, s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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