Polytope of Type {4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3}*48
Also Known As : cube, {4,3}. if this polytope has another name.
Group : SmallGroup(48,48)
Rank : 3
Schlafli Type : {4,3}
Number of vertices, edges, etc : 8, 12, 6
Order of s0s1s2 : 6
Order of s0s1s2s1 : 4
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,3,2} of size 96
   {4,3,3} of size 192
   {4,3,4} of size 192
   {4,3,6} of size 288
   {4,3,3} of size 384
   {4,3,6} of size 384
   {4,3,6} of size 384
   {4,3,4} of size 384
   {4,3,12} of size 768
   {4,3,6} of size 768
   {4,3,12} of size 768
   {4,3,8} of size 768
   {4,3,8} of size 768
   {4,3,4} of size 768
   {4,3,6} of size 864
   {4,3,6} of size 1152
   {4,3,6} of size 1152
   {4,3,12} of size 1152
   {4,3,4} of size 1296
   {4,3,6} of size 1296
   {4,3,5} of size 1920
Vertex Figure Of :
   {2,4,3} of size 96
   {4,4,3} of size 192
   {6,4,3} of size 288
   {4,4,3} of size 384
   {8,4,3} of size 384
   {10,4,3} of size 480
   {12,4,3} of size 576
   {3,4,3} of size 576
   {14,4,3} of size 672
   {4,4,3} of size 720
   {4,4,3} of size 768
   {4,4,3} of size 768
   {4,4,3} of size 768
   {16,4,3} of size 768
   {18,4,3} of size 864
   {20,4,3} of size 960
   {22,4,3} of size 1056
   {12,4,3} of size 1152
   {24,4,3} of size 1152
   {3,4,3} of size 1152
   {6,4,3} of size 1152
   {6,4,3} of size 1152
   {26,4,3} of size 1248
   {6,4,3} of size 1296
   {28,4,3} of size 1344
   {4,4,3} of size 1440
   {4,4,3} of size 1440
   {4,4,3} of size 1440
   {30,4,3} of size 1440
   {34,4,3} of size 1632
   {36,4,3} of size 1728
   {9,4,3} of size 1728
   {38,4,3} of size 1824
   {20,4,3} of size 1920
   {40,4,3} of size 1920
   {6,4,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3}*24
   4-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,3}*96, {4,6}*96
   3-fold covers : {4,9}*144, {12,3}*144
   4-fold covers : {8,3}*192, {4,12}*192b, {4,6}*192b, {4,12}*192c, {8,6}*192b, {8,6}*192c
   5-fold covers : {4,15}*240
   6-fold covers : {8,9}*288, {4,18}*288, {24,3}*288, {12,6}*288a, {12,6}*288b
   7-fold covers : {4,21}*336
   8-fold covers : {8,3}*384, {8,6}*384a, {4,12}*384d, {8,12}*384e, {8,12}*384f, {4,6}*384a, {8,6}*384d, {8,6}*384e, {8,6}*384f, {8,12}*384g, {8,12}*384h, {4,24}*384c, {4,24}*384d, {8,6}*384g, {4,12}*384e, {4,24}*384e, {4,6}*384b, {4,24}*384f
   9-fold covers : {4,27}*432, {12,9}*432, {12,3}*432
   10-fold covers : {8,15}*480, {20,6}*480c, {4,30}*480
   11-fold covers : {4,33}*528
   12-fold covers : {8,9}*576, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c, {24,3}*576, {12,12}*576f, {12,12}*576g, {12,6}*576b, {12,12}*576i, {24,6}*576b, {24,6}*576c, {24,6}*576d, {24,6}*576e, {12,6}*576f, {12,12}*576k, {12,3}*576
   13-fold covers : {4,39}*624
   14-fold covers : {8,21}*672, {28,6}*672, {4,42}*672
   15-fold covers : {4,45}*720, {12,15}*720
   16-fold covers : {16,3}*768a, {16,3}*768b, {8,6}*768d, {8,12}*768k, {8,6}*768e, {8,6}*768f, {8,12}*768l, {8,6}*768g, {8,6}*768h, {8,6}*768i, {8,12}*768m, {8,12}*768n, {8,24}*768i, {8,24}*768j, {8,24}*768k, {8,24}*768l, {8,6}*768j, {8,24}*768m, {8,12}*768o, {8,24}*768n, {8,12}*768p, {8,24}*768o, {8,24}*768p, {4,12}*768b, {4,6}*768a, {4,12}*768c, {8,12}*768q, {8,12}*768r, {8,12}*768s, {4,24}*768i, {4,12}*768d, {8,12}*768t, {4,24}*768j, {8,12}*768u, {4,12}*768e, {4,24}*768k, {8,6}*768k, {8,12}*768v, {8,12}*768w, {4,12}*768f, {4,24}*768l, {8,6}*768l, {8,12}*768x, {8,6}*768m, {8,6}*768n, {4,6}*768b, {4,6}*768c, {4,12}*768g, {4,12}*768h, {4,48}*768c, {4,48}*768d, {16,6}*768b, {16,6}*768c
   17-fold covers : {4,51}*816
   18-fold covers : {8,27}*864, {4,54}*864, {24,9}*864, {24,3}*864, {36,6}*864, {12,18}*864a, {12,18}*864b, {12,6}*864a, {12,6}*864b, {12,6}*864c
   19-fold covers : {4,57}*912
   20-fold covers : {8,15}*960a, {20,12}*960b, {20,6}*960e, {40,6}*960d, {40,6}*960e, {20,12}*960c, {4,60}*960b, {4,30}*960b, {4,60}*960c, {8,30}*960b, {8,30}*960c
   21-fold covers : {4,63}*1008, {12,21}*1008
   22-fold covers : {8,33}*1056, {44,6}*1056, {4,66}*1056
   23-fold covers : {4,69}*1104
   24-fold covers : {8,9}*1152, {8,18}*1152a, {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,18}*1152a, {8,18}*1152d, {8,18}*1152e, {8,18}*1152f, {8,36}*1152g, {8,36}*1152h, {4,72}*1152c, {4,72}*1152d, {8,18}*1152g, {4,36}*1152e, {4,72}*1152e, {4,18}*1152b, {4,72}*1152f, {24,3}*1152a, {24,6}*1152a, {24,6}*1152b, {24,6}*1152c, {24,12}*1152i, {24,12}*1152j, {24,12}*1152k, {24,12}*1152l, {24,12}*1152m, {24,6}*1152d, {24,12}*1152n, {12,6}*1152b, {12,6}*1152c, {24,6}*1152e, {24,6}*1152f, {12,24}*1152o, {12,24}*1152p, {12,24}*1152q, {12,24}*1152r, {24,6}*1152h, {12,6}*1152d, {12,24}*1152s, {12,12}*1152i, {12,24}*1152t, {12,12}*1152n, {12,12}*1152o, {24,6}*1152k, {24,6}*1152l, {24,12}*1152u, {24,12}*1152v, {12,12}*1152r, {12,24}*1152w, {12,6}*1152f, {12,24}*1152x, {12,3}*1152b, {12,6}*1152g, {24,3}*1152b, {24,3}*1152c, {12,6}*1152j, {12,12}*1152t
   25-fold covers : {4,75}*1200, {20,15}*1200, {20,3}*1200
   26-fold covers : {8,39}*1248, {52,6}*1248, {4,78}*1248
   27-fold covers : {4,81}*1296, {12,27}*1296, {36,9}*1296, {36,3}*1296, {12,3}*1296a, {12,9}*1296a, {12,9}*1296b, {12,9}*1296c, {12,9}*1296d, {4,9}*1296a, {4,9}*1296b, {12,3}*1296b, {12,9}*1296e, {12,9}*1296f
   28-fold covers : {8,21}*1344, {28,12}*1344b, {28,6}*1344e, {56,6}*1344b, {56,6}*1344c, {28,12}*1344c, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
   29-fold covers : {4,87}*1392
   30-fold covers : {8,45}*1440, {20,18}*1440, {4,90}*1440, {24,15}*1440, {60,6}*1440c, {12,30}*1440a, {12,30}*1440b, {60,6}*1440d
   31-fold covers : {4,93}*1488
   33-fold covers : {4,99}*1584, {12,33}*1584
   34-fold covers : {8,51}*1632, {68,6}*1632, {4,102}*1632
   35-fold covers : {4,105}*1680
   36-fold covers : {8,27}*1728, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c, {24,9}*1728, {24,3}*1728, {36,12}*1728c, {36,6}*1728b, {72,6}*1728b, {72,6}*1728c, {36,12}*1728d, {12,36}*1728e, {12,36}*1728f, {12,18}*1728c, {12,36}*1728g, {12,12}*1728k, {12,12}*1728l, {12,6}*1728b, {12,12}*1728n, {24,18}*1728b, {24,18}*1728c, {24,18}*1728d, {24,6}*1728b, {24,6}*1728c, {24,6}*1728d, {24,18}*1728e, {24,6}*1728e, {12,18}*1728d, {12,36}*1728h, {12,6}*1728f, {12,12}*1728p, {12,9}*1728, {12,3}*1728, {24,6}*1728f, {24,6}*1728g, {12,12}*1728w, {12,6}*1728i, {12,12}*1728y, {4,6}*1728, {4,12}*1728e, {12,12}*1728ab
   37-fold covers : {4,111}*1776
   38-fold covers : {8,57}*1824, {76,6}*1824, {4,114}*1824
   39-fold covers : {4,117}*1872, {12,39}*1872
   40-fold covers : {8,15}*1920a, {8,30}*1920a, {40,6}*1920a, {40,12}*1920e, {40,12}*1920f, {40,6}*1920b, {20,6}*1920a, {40,6}*1920c, {20,24}*1920c, {20,24}*1920d, {40,6}*1920d, {20,6}*1920b, {20,12}*1920b, {20,12}*1920c, {40,12}*1920g, {40,12}*1920h, {20,24}*1920e, {20,24}*1920f, {4,60}*1920d, {8,60}*1920e, {8,60}*1920f, {4,30}*1920a, {8,30}*1920d, {8,30}*1920e, {8,30}*1920f, {8,60}*1920g, {8,60}*1920h, {4,120}*1920c, {4,120}*1920d, {8,30}*1920g, {4,60}*1920e, {4,120}*1920e, {4,30}*1920b, {4,120}*1920f
   41-fold covers : {4,123}*1968
Permutation Representation (GAP) : 
s0 := (4,6);;
s1 := (3,4)(5,6);;
s2 := (1,3)(2,5);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(6)!(4,6);
s1 := Sym(6)!(3,4)(5,6);
s2 := Sym(6)!(1,3)(2,5);
poly := sub<Sym(6)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope