Polytope of Type {3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4}*24
Also Known As : hemioctahedron, {3,4}₃. if this polytope has another name.
Group : SmallGroup(24,12)
Rank : 3
Schlafli Type : {3,4}
Number of vertices, edges, etc : 3, 6, 4
Order of s₀s₁s₂ : 3
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Projective
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {3,4,2} of size 48
   {3,4,4} of size 192
   {3,4,4} of size 384
   {3,4,6} of size 720
   {3,4,8} of size 768
   {3,4,8} of size 768
   {3,4,6} of size 1296
Vertex Figure Of :
   {2,3,4} of size 48
   {4,3,4} of size 96
   {6,3,4} of size 144
   {4,3,4} of size 192
   {8,3,4} of size 384
   {6,3,4} of size 432
   {6,3,4} of size 576
   {12,3,4} of size 576
   {8,3,4} of size 768
   {12,3,4} of size 1152
   {24,3,4} of size 1152
   {10,3,4} of size 1200
   {18,3,4} of size 1296
   {6,3,4} of size 1296
   {10,3,4} of size 1440
   {6,3,4} of size 1728
   {12,3,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,4}*48, {6,4}*48b, {6,4}*48c
   3-fold covers : {9,4}*72
   4-fold covers : {12,4}*96b, {12,4}*96c, {3,8}*96, {6,4}*96
   5-fold covers : {15,4}*120
   6-fold covers : {9,4}*144, {18,4}*144b, {18,4}*144c, {3,12}*144, {6,12}*144d
   7-fold covers : {21,4}*168
   8-fold covers : {6,4}*192a, {3,8}*192, {6,8}*192a, {24,4}*192c, {24,4}*192d, {12,4}*192b, {6,4}*192b, {12,4}*192c, {6,8}*192b, {6,8}*192c
   9-fold covers : {27,4}*216
   10-fold covers : {6,20}*240b, {15,4}*240, {30,4}*240b, {30,4}*240c
   11-fold covers : {33,4}*264
   12-fold covers : {36,4}*288b, {36,4}*288c, {9,8}*288, {18,4}*288, {3,24}*288, {6,12}*288a, {6,12}*288b
   13-fold covers : {39,4}*312
   14-fold covers : {6,28}*336b, {21,4}*336, {42,4}*336b, {42,4}*336c
   15-fold covers : {45,4}*360
   16-fold covers : {12,4}*384b, {12,4}*384c, {3,8}*384, {6,8}*384a, {12,8}*384c, {12,8}*384d, {6,8}*384b, {6,8}*384c, {48,4}*384c, {48,4}*384d, {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
   17-fold covers : {51,4}*408
   18-fold covers : {27,4}*432, {54,4}*432b, {54,4}*432c, {6,36}*432c, {9,12}*432, {18,12}*432c, {3,12}*432, {6,12}*432d
   19-fold covers : {57,4}*456
   20-fold covers : {60,4}*480b, {60,4}*480c, {15,8}*480, {6,20}*480c, {30,4}*480
   21-fold covers : {63,4}*504
   22-fold covers : {6,44}*528b, {33,4}*528, {66,4}*528b, {66,4}*528c
   23-fold covers : {69,4}*552
   24-fold covers : {18,4}*576a, {9,8}*576, {18,8}*576a, {72,4}*576c, {72,4}*576d, {36,4}*576b, {18,4}*576b, {36,4}*576c, {18,8}*576b, {18,8}*576c, {3,24}*576, {6,24}*576a, {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, {12,12}*576l
   25-fold covers : {75,4}*600
   26-fold covers : {6,52}*624b, {39,4}*624, {78,4}*624b, {78,4}*624c
   27-fold covers : {81,4}*648, {3,12}*648, {9,4}*648, {9,12}*648a, {9,12}*648b
   28-fold covers : {84,4}*672b, {84,4}*672c, {21,8}*672, {6,28}*672, {42,4}*672
   29-fold covers : {87,4}*696
   30-fold covers : {18,20}*720b, {45,4}*720, {90,4}*720b, {90,4}*720c, {15,12}*720, {30,12}*720d, {6,60}*720d
   31-fold covers : {93,4}*744
   32-fold covers : {24,4}*768e, {24,4}*768f, {3,16}*768a, {3,16}*768b, {6,16}*768a, {12,8}*768e, {12,8}*768f, {12,8}*768g, {12,8}*768h, {24,4}*768g, {24,4}*768h, {6,8}*768a, {6,8}*768b, {6,8}*768c, {12,8}*768i, {12,8}*768j, {96,4}*768c, {96,4}*768d, {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
   33-fold covers : {99,4}*792
   34-fold covers : {6,68}*816b, {51,4}*816, {102,4}*816b, {102,4}*816c
   35-fold covers : {105,4}*840
   36-fold covers : {108,4}*864b, {108,4}*864c, {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, {12,12}*864n, {6,12}*864c
   37-fold covers : {111,4}*888
   38-fold covers : {6,76}*912b, {57,4}*912, {114,4}*912b, {114,4}*912c
   39-fold covers : {117,4}*936
   40-fold covers : {6,40}*960c, {30,4}*960a, {15,8}*960a, {30,8}*960a, {120,4}*960c, {120,4}*960d, {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
   41-fold covers : {123,4}*984
   42-fold covers : {18,28}*1008b, {63,4}*1008, {126,4}*1008b, {126,4}*1008c, {21,12}*1008, {42,12}*1008d, {6,84}*1008d
   43-fold covers : {129,4}*1032
   44-fold covers : {132,4}*1056b, {132,4}*1056c, {33,8}*1056, {6,44}*1056, {66,4}*1056
   45-fold covers : {135,4}*1080
   46-fold covers : {6,92}*1104b, {69,4}*1104, {138,4}*1104b, {138,4}*1104c
   47-fold covers : {141,4}*1128
   48-fold covers : {36,4}*1152b, {36,4}*1152c, {9,8}*1152, {18,8}*1152a, {36,8}*1152c, {36,8}*1152d, {18,8}*1152b, {18,8}*1152c, {144,4}*1152c, {144,4}*1152d, {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, {12,24}*1152g, {12,24}*1152h, {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, {12,24}*1152y, {12,24}*1152z, {24,12}*1152y, {24,12}*1152z, {3,24}*1152c, {6,12}*1152j, {12,12}*1152t
   49-fold covers : {147,4}*1176
   50-fold covers : {6,100}*1200b, {75,4}*1200, {150,4}*1200b, {150,4}*1200c, {15,20}*1200, {30,20}*1200d, {3,20}*1200, {6,20}*1200d
   51-fold covers : {153,4}*1224
   52-fold covers : {156,4}*1248b, {156,4}*1248c, {39,8}*1248, {6,52}*1248, {78,4}*1248
   53-fold covers : {159,4}*1272
   54-fold covers : {81,4}*1296, {162,4}*1296b, {162,4}*1296c, {6,108}*1296c, {27,12}*1296, {54,12}*1296c, {9,36}*1296, {18,36}*1296d, {6,36}*1296i, {3,36}*1296, {6,36}*1296j, {6,36}*1296k, {3,12}*1296a, {18,12}*1296i, {9,12}*1296a, {18,12}*1296j, {6,12}*1296e, {9,12}*1296b, {9,12}*1296c, {18,12}*1296k, {6,12}*1296f, {9,12}*1296d, {6,4}*1296b, {6,4}*1296c, {6,12}*1296p, {9,4}*1296a, {3,12}*1296b, {6,12}*1296q, {6,12}*1296r, {9,4}*1296b, {9,12}*1296e, {9,12}*1296f, {18,4}*1296c, {18,4}*1296d, {18,12}*1296m, {18,12}*1296n, {18,12}*1296o, {18,12}*1296p
   55-fold covers : {165,4}*1320
   56-fold covers : {6,56}*1344a, {42,4}*1344a, {21,8}*1344, {42,8}*1344a, {168,4}*1344c, {168,4}*1344d, {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
   57-fold covers : {171,4}*1368
   58-fold covers : {6,116}*1392b, {87,4}*1392, {174,4}*1392b, {174,4}*1392c
   59-fold covers : {177,4}*1416
   60-fold covers : {180,4}*1440b, {180,4}*1440c, {45,8}*1440, {18,20}*1440, {90,4}*1440, {15,24}*1440, {3,20}*1440b, {15,12}*1440d, {15,20}*1440b, {6,60}*1440c, {30,12}*1440a, {30,12}*1440b, {6,60}*1440d
   61-fold covers : {183,4}*1464
   62-fold covers : {6,124}*1488b, {93,4}*1488, {186,4}*1488b, {186,4}*1488c
   63-fold covers : {189,4}*1512
   65-fold covers : {195,4}*1560
   66-fold covers : {18,44}*1584b, {99,4}*1584, {198,4}*1584b, {198,4}*1584c, {33,12}*1584, {66,12}*1584d, {6,132}*1584d
   67-fold covers : {201,4}*1608
   68-fold covers : {204,4}*1632b, {204,4}*1632c, {51,8}*1632, {6,68}*1632, {102,4}*1632
   69-fold covers : {207,4}*1656
   70-fold covers : {30,28}*1680b, {42,20}*1680b, {6,140}*1680b, {105,4}*1680, {210,4}*1680b, {210,4}*1680c
   71-fold covers : {213,4}*1704
   72-fold covers : {54,4}*1728a, {27,8}*1728, {54,8}*1728a, {216,4}*1728c, {216,4}*1728d, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c, {6,72}*1728a, {9,24}*1728, {18,24}*1728a, {3,24}*1728, {6,24}*1728a, {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, {12,36}*1728i, {36,12}*1728i, {3,12}*1728, {12,12}*1728u, {6,24}*1728f, {6,24}*1728g, {12,12}*1728v, {6,12}*1728i, {12,12}*1728x, {6,4}*1728, {12,4}*1728e, {12,12}*1728aa
   73-fold covers : {219,4}*1752
   74-fold covers : {6,148}*1776b, {111,4}*1776, {222,4}*1776b, {222,4}*1776c
   75-fold covers : {225,4}*1800
   76-fold covers : {228,4}*1824b, {228,4}*1824c, {57,8}*1824, {6,76}*1824, {114,4}*1824
   77-fold covers : {231,4}*1848
   78-fold covers : {18,52}*1872b, {117,4}*1872, {234,4}*1872b, {234,4}*1872c, {39,12}*1872, {78,12}*1872d, {6,156}*1872d
   79-fold covers : {237,4}*1896
   80-fold covers : {12,40}*1920c, {12,40}*1920d, {60,4}*1920b, {60,4}*1920c, {15,8}*1920a, {30,8}*1920a, {60,8}*1920c, {60,8}*1920d, {30,8}*1920b, {30,8}*1920c, {240,4}*1920c, {240,4}*1920d, {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, {15,4}*1920b
   81-fold covers : {243,4}*1944, {9,4}*1944, {9,12}*1944a, {9,12}*1944b, {9,12}*1944c
   82-fold covers : {6,164}*1968b, {123,4}*1968, {246,4}*1968b, {246,4}*1968c
   83-fold covers : {249,4}*1992
Permutation Representation (GAP) :

s0 := (3,4);;
s1 := (2,3);;
s2 := (1,2)(3,4);;
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, s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(4)!(3,4);
s1 := Sym(4)!(2,3);
s2 := Sym(4)!(1,2)(3,4);
poly := sub<Sym(4)|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, 
s0*s2*s1*s0*s2*s1*s0*s2*s1 >;

References : None.
to this polytope