Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*48
Also Known As : {3,6}(2,0), {3,6}4if this polytope has another name.
Group : SmallGroup(48,48)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 4, 12, 8
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,6,2} of size 96
   {3,6,4} of size 192
   {3,6,3} of size 240
   {3,6,6} of size 288
   {3,6,4} of size 384
   {3,6,4} of size 384
   {3,6,8} of size 384
   {3,6,6} of size 480
   {3,6,10} of size 480
   {3,6,12} of size 576
   {3,6,14} of size 672
   {3,6,4} of size 720
   {3,6,3} of size 720
   {3,6,4} of size 768
   {3,6,4} of size 768
   {3,6,4} of size 768
   {3,6,16} of size 768
   {3,6,18} of size 864
   {3,6,12} of size 960
   {3,6,20} of size 960
   {3,6,22} of size 1056
   {3,6,12} of size 1152
   {3,6,24} of size 1152
   {3,6,15} of size 1200
   {3,6,26} of size 1248
   {3,6,6} of size 1296
   {3,6,28} of size 1344
   {3,6,4} of size 1440
   {3,6,4} of size 1440
   {3,6,4} of size 1440
   {3,6,6} of size 1440
   {3,6,6} of size 1440
   {3,6,30} of size 1440
   {3,6,34} of size 1632
   {3,6,21} of size 1680
   {3,6,36} of size 1728
   {3,6,38} of size 1824
   {3,6,20} of size 1920
   {3,6,40} of size 1920
   {3,6,24} of size 1920
   {3,6,6} of size 1920
Vertex Figure Of :
   {2,3,6} of size 96
   {3,3,6} of size 240
   {4,3,6} of size 384
   {4,3,6} of size 384
   {6,3,6} of size 480
   {4,3,6} of size 768
   {6,3,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12}*96, {6,6}*96
   3-fold covers : {3,6}*144
   4-fold covers : {3,6}*192, {6,12}*192a, {12,6}*192a, {6,12}*192b, {12,6}*192b, {6,6}*192b
   5-fold covers : {15,6}*240
   6-fold covers : {3,12}*288, {6,6}*288a, {6,6}*288b
   7-fold covers : {21,6}*336
   8-fold covers : {3,12}*384, {12,12}*384a, {12,12}*384b, {6,6}*384c, {6,6}*384d, {6,6}*384e, {6,12}*384, {12,6}*384, {12,12}*384c, {12,12}*384d, {6,24}*384a, {24,6}*384a, {6,24}*384b, {24,6}*384b
   9-fold covers : {9,6}*432, {3,6}*432
   10-fold covers : {15,12}*480, {6,30}*480, {30,6}*480
   11-fold covers : {33,6}*528
   12-fold covers : {3,6}*576, {6,12}*576a, {12,6}*576a, {6,12}*576c, {12,6}*576c, {6,6}*576a, {6,6}*576b, {6,12}*576d, {12,6}*576d, {6,12}*576e, {12,6}*576e, {3,12}*576
   13-fold covers : {39,6}*624
   14-fold covers : {21,12}*672, {6,42}*672, {42,6}*672
   15-fold covers : {15,6}*720e
   16-fold covers : {3,24}*768, {3,6}*768, {6,12}*768c, {12,6}*768c, {6,12}*768d, {12,6}*768d, {6,12}*768e, {12,6}*768e, {6,6}*768b, {6,6}*768c, {6,6}*768d, {6,24}*768, {24,6}*768, {12,24}*768a, {24,12}*768a, {12,24}*768b, {24,12}*768b, {6,12}*768f, {12,6}*768f, {12,12}*768a, {12,12}*768b, {12,12}*768c, {12,24}*768c, {24,12}*768c, {12,24}*768d, {24,12}*768d, {6,12}*768g, {12,6}*768g, {12,24}*768e, {24,12}*768e, {12,24}*768f, {24,12}*768f, {6,12}*768h, {12,6}*768h, {6,6}*768e, {6,12}*768i, {12,6}*768i, {6,6}*768f, {6,12}*768j, {12,6}*768j, {6,48}*768a, {48,6}*768a, {6,48}*768b, {48,6}*768b
   17-fold covers : {51,6}*816
   18-fold covers : {9,12}*864, {3,12}*864, {6,18}*864, {18,6}*864, {6,6}*864a, {6,6}*864b, {6,6}*864c
   19-fold covers : {57,6}*912
   20-fold covers : {15,6}*960, {6,60}*960a, {60,6}*960a, {12,30}*960a, {30,12}*960a, {6,30}*960, {30,6}*960, {6,60}*960b, {60,6}*960b, {12,30}*960b, {30,12}*960b
   21-fold covers : {21,6}*1008b
   22-fold covers : {33,12}*1056, {6,66}*1056, {66,6}*1056
   23-fold covers : {69,6}*1104
   24-fold covers : {3,12}*1152a, {6,6}*1152a, {6,6}*1152b, {12,12}*1152d, {12,12}*1152e, {12,12}*1152f, {12,12}*1152g, {6,12}*1152a, {12,6}*1152a, {6,6}*1152c, {6,6}*1152d, {6,6}*1152e, {6,6}*1152f, {6,24}*1152g, {24,6}*1152g, {6,24}*1152i, {24,6}*1152i, {12,12}*1152j, {12,12}*1152l, {6,24}*1152j, {24,6}*1152j, {6,12}*1152e, {12,6}*1152e, {12,12}*1152p, {12,12}*1152q, {6,24}*1152m, {24,6}*1152m, {3,12}*1152b, {3,24}*1152b, {6,12}*1152g, {3,24}*1152c, {6,12}*1152j, {12,6}*1152j
   25-fold covers : {75,6}*1200, {15,30}*1200, {3,6}*1200
   26-fold covers : {39,12}*1248, {6,78}*1248, {78,6}*1248
   27-fold covers : {27,6}*1296, {9,18}*1296a, {9,6}*1296a, {3,6}*1296, {9,6}*1296b, {3,18}*1296a, {9,6}*1296c, {9,6}*1296d, {9,6}*1296e, {3,18}*1296b, {9,6}*1296f, {9,18}*1296b, {9,18}*1296c
   28-fold covers : {21,6}*1344, {6,84}*1344a, {84,6}*1344a, {12,42}*1344a, {42,12}*1344a, {6,42}*1344, {42,6}*1344, {6,84}*1344b, {84,6}*1344b, {12,42}*1344b, {42,12}*1344b
   29-fold covers : {87,6}*1392
   30-fold covers : {15,12}*1440c, {6,30}*1440g, {30,6}*1440g, {6,30}*1440h, {30,6}*1440h
   31-fold covers : {93,6}*1488
   33-fold covers : {33,6}*1584
   34-fold covers : {51,12}*1632, {6,102}*1632, {102,6}*1632
   35-fold covers : {105,6}*1680
   36-fold covers : {9,6}*1728, {3,6}*1728, {6,36}*1728a, {36,6}*1728a, {12,18}*1728a, {18,12}*1728a, {6,18}*1728a, {18,6}*1728a, {6,36}*1728c, {36,6}*1728c, {12,18}*1728b, {18,12}*1728b, {6,12}*1728a, {12,6}*1728a, {6,12}*1728c, {12,6}*1728c, {6,6}*1728a, {6,6}*1728b, {6,12}*1728d, {12,6}*1728d, {6,12}*1728e, {12,6}*1728e, {9,12}*1728, {3,12}*1728, {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h, {6,12}*1728j, {12,6}*1728j, {12,12}*1728z
   37-fold covers : {111,6}*1776
   38-fold covers : {57,12}*1824, {6,114}*1824, {114,6}*1824
   39-fold covers : {39,6}*1872
   40-fold covers : {15,12}*1920, {6,30}*1920a, {30,6}*1920a, {12,60}*1920a, {60,12}*1920a, {12,60}*1920b, {60,12}*1920b, {6,60}*1920, {60,6}*1920, {6,30}*1920b, {30,6}*1920b, {6,30}*1920c, {30,6}*1920c, {6,120}*1920a, {120,6}*1920a, {6,120}*1920b, {120,6}*1920b, {12,60}*1920c, {60,12}*1920c, {24,30}*1920a, {30,24}*1920a, {12,30}*1920, {30,12}*1920, {12,60}*1920d, {60,12}*1920d, {24,30}*1920b, {30,24}*1920b
   41-fold covers : {123,6}*1968
Permutation Representation (GAP) :
s0 := (1,4)(2,6);;
s1 := (3,4)(5,6);;
s2 := (1,4)(2,6)(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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(6)!(1,4)(2,6);
s1 := Sym(6)!(3,4)(5,6);
s2 := Sym(6)!(1,4)(2,6)(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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >; 
 
References : None.
to this polytope