Polytope of Type {3,2,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4}*48
if this polytope has a name.
Group : SmallGroup(48,38)
Rank : 4
Schlafli Type : {3,2,4}
Number of vertices, edges, etc : 3, 3, 4, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Projective
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,2,4,2} of size 96
{3,2,4,3} of size 144
{3,2,4,4} of size 192
{3,2,4,6} of size 288
{3,2,4,3} of size 288
{3,2,4,6} of size 288
{3,2,4,6} of size 288
{3,2,4,8} of size 384
{3,2,4,8} of size 384
{3,2,4,4} of size 384
{3,2,4,9} of size 432
{3,2,4,4} of size 432
{3,2,4,6} of size 432
{3,2,4,10} of size 480
{3,2,4,12} of size 576
{3,2,4,12} of size 576
{3,2,4,12} of size 576
{3,2,4,6} of size 576
{3,2,4,14} of size 672
{3,2,4,5} of size 720
{3,2,4,6} of size 720
{3,2,4,15} of size 720
{3,2,4,8} of size 768
{3,2,4,16} of size 768
{3,2,4,16} of size 768
{3,2,4,4} of size 768
{3,2,4,8} of size 768
{3,2,4,18} of size 864
{3,2,4,9} of size 864
{3,2,4,18} of size 864
{3,2,4,18} of size 864
{3,2,4,4} of size 864
{3,2,4,6} of size 864
{3,2,4,20} of size 960
{3,2,4,5} of size 960
{3,2,4,21} of size 1008
{3,2,4,22} of size 1056
{3,2,4,24} of size 1152
{3,2,4,24} of size 1152
{3,2,4,12} of size 1152
{3,2,4,6} of size 1152
{3,2,4,24} of size 1152
{3,2,4,24} of size 1152
{3,2,4,12} of size 1152
{3,2,4,6} of size 1152
{3,2,4,12} of size 1152
{3,2,4,4} of size 1200
{3,2,4,10} of size 1200
{3,2,4,26} of size 1248
{3,2,4,27} of size 1296
{3,2,4,6} of size 1296
{3,2,4,12} of size 1296
{3,2,4,28} of size 1344
{3,2,4,30} of size 1440
{3,2,4,5} of size 1440
{3,2,4,6} of size 1440
{3,2,4,6} of size 1440
{3,2,4,6} of size 1440
{3,2,4,10} of size 1440
{3,2,4,10} of size 1440
{3,2,4,15} of size 1440
{3,2,4,30} of size 1440
{3,2,4,30} of size 1440
{3,2,4,33} of size 1584
{3,2,4,34} of size 1632
{3,2,4,36} of size 1728
{3,2,4,36} of size 1728
{3,2,4,36} of size 1728
{3,2,4,18} of size 1728
{3,2,4,4} of size 1728
{3,2,4,12} of size 1728
{3,2,4,38} of size 1824
{3,2,4,39} of size 1872
{3,2,4,40} of size 1920
{3,2,4,40} of size 1920
{3,2,4,20} of size 1920
{3,2,4,5} of size 1920
{3,2,4,10} of size 1920
{3,2,4,10} of size 1920
Vertex Figure Of :
{2,3,2,4} of size 96
{3,3,2,4} of size 192
{4,3,2,4} of size 192
{6,3,2,4} of size 288
{4,3,2,4} of size 384
{6,3,2,4} of size 384
{5,3,2,4} of size 480
{8,3,2,4} of size 768
{12,3,2,4} of size 768
{6,3,2,4} of size 864
{5,3,2,4} of size 960
{10,3,2,4} of size 960
{10,3,2,4} of size 960
{6,3,2,4} of size 1152
{12,3,2,4} of size 1152
{10,3,2,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,2,8}*96, {6,2,4}*96
3-fold covers : {9,2,4}*144, {3,2,12}*144, {3,6,4}*144
4-fold covers : {3,2,16}*192, {12,2,4}*192, {6,4,4}*192, {6,2,8}*192, {3,4,4}*192b
5-fold covers : {3,2,20}*240, {15,2,4}*240
6-fold covers : {9,2,8}*288, {18,2,4}*288, {3,2,24}*288, {3,6,8}*288, {6,2,12}*288, {6,6,4}*288a, {6,6,4}*288c
7-fold covers : {3,2,28}*336, {21,2,4}*336
8-fold covers : {3,2,32}*384, {12,4,4}*384, {24,2,4}*384, {12,2,8}*384, {6,4,8}*384a, {6,8,4}*384a, {6,4,8}*384b, {6,8,4}*384b, {6,4,4}*384a, {6,2,16}*384, {3,8,4}*384, {3,4,8}*384, {6,4,4}*384d
9-fold covers : {27,2,4}*432, {3,2,36}*432, {9,2,12}*432, {3,6,12}*432a, {9,6,4}*432, {3,6,4}*432a, {3,6,12}*432b, {3,6,4}*432b
10-fold covers : {3,2,40}*480, {15,2,8}*480, {6,2,20}*480, {6,10,4}*480, {30,2,4}*480
11-fold covers : {3,2,44}*528, {33,2,4}*528
12-fold covers : {9,2,16}*576, {36,2,4}*576, {18,4,4}*576, {18,2,8}*576, {3,2,48}*576, {3,6,16}*576, {9,4,4}*576b, {12,2,12}*576, {6,4,12}*576, {6,12,4}*576a, {12,6,4}*576a, {6,2,24}*576, {6,6,8}*576a, {12,6,4}*576b, {6,6,8}*576c, {6,12,4}*576c, {3,4,12}*576, {3,6,4}*576a, {3,12,4}*576
13-fold covers : {3,2,52}*624, {39,2,4}*624
14-fold covers : {3,2,56}*672, {21,2,8}*672, {6,2,28}*672, {6,14,4}*672, {42,2,4}*672
15-fold covers : {9,2,20}*720, {45,2,4}*720, {3,6,20}*720, {15,2,12}*720, {3,2,60}*720, {15,6,4}*720
16-fold covers : {3,2,64}*768, {6,4,8}*768a, {6,8,4}*768a, {6,8,8}*768a, {6,8,8}*768b, {6,8,8}*768c, {6,8,8}*768d, {24,2,8}*768, {12,4,8}*768a, {24,4,4}*768a, {12,4,8}*768b, {24,4,4}*768b, {12,8,4}*768a, {12,4,4}*768a, {12,4,4}*768b, {12,8,4}*768b, {12,8,4}*768c, {12,8,4}*768d, {6,4,16}*768a, {6,16,4}*768a, {6,4,16}*768b, {6,16,4}*768b, {6,4,4}*768a, {6,4,8}*768b, {6,8,4}*768b, {12,2,16}*768, {48,2,4}*768, {6,2,32}*768, {3,8,8}*768, {3,4,4}*768a, {3,8,4}*768c, {3,8,4}*768d, {3,4,16}*768, {6,4,4}*768e, {12,4,4}*768e, {12,4,4}*768f, {6,8,4}*768c, {6,4,8}*768c, {6,8,4}*768d
17-fold covers : {3,2,68}*816, {51,2,4}*816
18-fold covers : {27,2,8}*864, {54,2,4}*864, {3,2,72}*864, {9,2,24}*864, {3,6,24}*864a, {9,6,8}*864, {3,6,8}*864a, {6,2,36}*864, {18,2,12}*864, {6,6,12}*864a, {6,18,4}*864a, {18,6,4}*864a, {6,6,4}*864b, {18,6,4}*864b, {6,6,4}*864c, {3,6,24}*864b, {3,6,8}*864b, {6,6,12}*864b, {6,6,12}*864c, {6,6,12}*864e, {6,6,4}*864h, {6,6,12}*864f, {6,6,4}*864j, {6,6,4}*864k
19-fold covers : {3,2,76}*912, {57,2,4}*912
20-fold covers : {3,2,80}*960, {15,2,16}*960, {12,2,20}*960, {6,4,20}*960, {6,20,4}*960, {12,10,4}*960, {6,2,40}*960, {6,10,8}*960, {60,2,4}*960, {30,4,4}*960, {30,2,8}*960, {3,4,20}*960, {15,4,4}*960b
21-fold covers : {9,2,28}*1008, {63,2,4}*1008, {3,6,28}*1008, {21,2,12}*1008, {3,2,84}*1008, {21,6,4}*1008
22-fold covers : {3,2,88}*1056, {33,2,8}*1056, {6,2,44}*1056, {6,22,4}*1056, {66,2,4}*1056
23-fold covers : {3,2,92}*1104, {69,2,4}*1104
24-fold covers : {9,2,32}*1152, {3,6,32}*1152, {3,2,96}*1152, {36,4,4}*1152, {12,12,4}*1152b, {12,12,4}*1152c, {12,4,12}*1152, {18,4,8}*1152a, {18,8,4}*1152a, {6,8,12}*1152a, {6,12,8}*1152b, {6,12,8}*1152c, {6,24,4}*1152a, {6,4,24}*1152a, {6,24,4}*1152c, {18,4,8}*1152b, {18,8,4}*1152b, {6,8,12}*1152b, {6,12,8}*1152e, {6,12,8}*1152f, {6,24,4}*1152d, {6,4,24}*1152b, {6,24,4}*1152f, {18,4,4}*1152a, {6,4,12}*1152a, {6,12,4}*1152b, {6,12,4}*1152c, {36,2,8}*1152, {72,2,4}*1152, {12,6,8}*1152b, {12,6,8}*1152c, {24,6,4}*1152b, {24,6,4}*1152c, {12,2,24}*1152, {24,2,12}*1152, {18,2,16}*1152, {6,6,16}*1152b, {6,6,16}*1152c, {6,2,48}*1152, {9,8,4}*1152, {9,4,8}*1152, {18,4,4}*1152d, {3,8,12}*1152, {3,4,24}*1152, {3,6,8}*1152, {3,12,8}*1152, {3,12,4}*1152b, {3,24,4}*1152, {6,4,12}*1152b, {6,4,12}*1152c, {6,6,4}*1152c, {6,6,12}*1152b, {6,12,4}*1152i, {12,6,4}*1152b, {6,6,4}*1152f, {6,12,4}*1152j
25-fold covers : {3,2,100}*1200, {75,2,4}*1200, {3,10,4}*1200, {15,2,20}*1200, {15,10,4}*1200
26-fold covers : {3,2,104}*1248, {39,2,8}*1248, {6,2,52}*1248, {6,26,4}*1248, {78,2,4}*1248
27-fold covers : {81,2,4}*1296, {9,2,36}*1296, {9,6,12}*1296a, {3,6,36}*1296a, {27,2,12}*1296, {3,2,108}*1296, {3,6,12}*1296a, {3,6,12}*1296b, {9,18,4}*1296, {9,6,4}*1296a, {27,6,4}*1296, {9,6,4}*1296b, {9,6,4}*1296c, {9,6,4}*1296d, {3,6,4}*1296a, {3,18,4}*1296, {3,6,36}*1296b, {9,6,12}*1296b, {3,6,12}*1296c, {3,6,12}*1296d, {3,6,12}*1296e, {9,6,4}*1296e, {3,6,4}*1296b, {3,6,12}*1296f
28-fold covers : {3,2,112}*1344, {21,2,16}*1344, {12,2,28}*1344, {12,14,4}*1344, {6,4,28}*1344, {6,28,4}*1344, {6,2,56}*1344, {6,14,8}*1344, {84,2,4}*1344, {42,4,4}*1344, {42,2,8}*1344, {3,4,28}*1344, {21,4,4}*1344b
29-fold covers : {3,2,116}*1392, {87,2,4}*1392
30-fold covers : {9,2,40}*1440, {45,2,8}*1440, {18,2,20}*1440, {18,10,4}*1440, {90,2,4}*1440, {3,6,40}*1440, {15,2,24}*1440, {3,2,120}*1440, {15,6,8}*1440, {6,10,12}*1440, {6,6,20}*1440a, {6,6,20}*1440c, {6,30,4}*1440a, {30,2,12}*1440, {6,2,60}*1440, {6,30,4}*1440b, {30,6,4}*1440b, {30,6,4}*1440c
31-fold covers : {3,2,124}*1488, {93,2,4}*1488
33-fold covers : {9,2,44}*1584, {99,2,4}*1584, {3,6,44}*1584, {33,2,12}*1584, {3,2,132}*1584, {33,6,4}*1584
34-fold covers : {3,2,136}*1632, {51,2,8}*1632, {6,2,68}*1632, {6,34,4}*1632, {102,2,4}*1632
35-fold covers : {15,2,28}*1680, {21,2,20}*1680, {3,2,140}*1680, {105,2,4}*1680
36-fold covers : {27,2,16}*1728, {108,2,4}*1728, {54,4,4}*1728, {54,2,8}*1728, {3,2,144}*1728, {9,2,48}*1728, {3,6,48}*1728a, {9,6,16}*1728, {3,6,16}*1728a, {27,4,4}*1728b, {12,2,36}*1728, {36,2,12}*1728, {12,6,12}*1728a, {36,6,4}*1728a, {12,18,4}*1728a, {18,4,12}*1728, {18,12,4}*1728a, {6,4,36}*1728, {6,36,4}*1728a, {12,6,4}*1728a, {6,12,4}*1728b, {6,12,12}*1728a, {6,2,72}*1728, {18,2,24}*1728, {6,6,24}*1728a, {6,18,8}*1728a, {18,6,8}*1728a, {6,6,8}*1728b, {36,6,4}*1728b, {12,6,4}*1728b, {18,6,8}*1728b, {6,6,8}*1728c, {18,12,4}*1728b, {6,12,4}*1728c, {3,6,48}*1728b, {3,4,36}*1728, {9,6,4}*1728a, {9,4,12}*1728, {3,12,12}*1728a, {9,12,4}*1728, {3,6,4}*1728a, {3,12,4}*1728a, {3,6,16}*1728b, {6,6,24}*1728b, {6,6,24}*1728c, {6,6,24}*1728e, {12,6,12}*1728b, {12,6,12}*1728d, {12,6,12}*1728e, {12,6,12}*1728f, {6,12,12}*1728b, {6,12,12}*1728c, {6,12,12}*1728e, {6,6,8}*1728e, {6,6,24}*1728f, {6,12,4}*1728j, {6,12,12}*1728g, {12,6,4}*1728h, {3,6,12}*1728, {3,12,12}*1728b, {6,6,8}*1728f, {12,6,4}*1728k, {12,6,4}*1728l, {6,6,8}*1728g, {6,4,4}*1728b, {6,4,4}*1728c, {6,4,12}*1728b, {6,12,4}*1728n, {6,12,4}*1728p, {12,4,4}*1728c, {12,6,4}*1728n, {3,12,4}*1728b
37-fold covers : {3,2,148}*1776, {111,2,4}*1776
38-fold covers : {3,2,152}*1824, {57,2,8}*1824, {6,2,76}*1824, {6,38,4}*1824, {114,2,4}*1824
39-fold covers : {9,2,52}*1872, {117,2,4}*1872, {3,6,52}*1872, {39,2,12}*1872, {3,2,156}*1872, {39,6,4}*1872
40-fold covers : {15,2,32}*1920, {3,2,160}*1920, {60,4,4}*1920, {12,20,4}*1920, {12,4,20}*1920, {30,4,8}*1920a, {30,8,4}*1920a, {6,8,20}*1920a, {6,20,8}*1920a, {6,4,40}*1920a, {6,40,4}*1920a, {30,4,8}*1920b, {30,8,4}*1920b, {6,8,20}*1920b, {6,20,8}*1920b, {6,4,40}*1920b, {6,40,4}*1920b, {30,4,4}*1920a, {6,4,20}*1920a, {6,20,4}*1920a, {60,2,8}*1920, {120,2,4}*1920, {12,10,8}*1920, {24,10,4}*1920, {12,2,40}*1920, {24,2,20}*1920, {30,2,16}*1920, {6,10,16}*1920, {6,2,80}*1920, {3,8,20}*1920, {3,4,40}*1920, {15,8,4}*1920, {15,4,8}*1920, {6,4,20}*1920b, {6,20,4}*1920c, {30,4,4}*1920d
41-fold covers : {3,2,164}*1968, {123,2,4}*1968
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := (5,6);;
s3 := (4,5)(6,7);;
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*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(7)!(2,3);
s1 := Sym(7)!(1,2);
s2 := Sym(7)!(5,6);
s3 := Sym(7)!(4,5)(6,7);
poly := sub<Sym(7)|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*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope