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Polytope of Type {3,4}

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