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# Polytope of Type {3,3}

Atlas Canonical Name : {3,3}*24
Also Known As : tetrahedron, 3-simplex, {3,3}. if this polytope has another name.
Group : SmallGroup(24,12)
Rank : 3
Schlafli Type : {3,3}
Number of vertices, edges, etc : 4, 6, 4
Order of s0s1s2 : 4
Order of s0s1s2s1 : 3
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,3,2} of size 48
{3,3,3} of size 120
{3,3,4} of size 192
{3,3,6} of size 240
{3,3,4} of size 384
{3,3,6} of size 720
{3,3,8} of size 768
{3,3,8} of size 768
{3,3,6} of size 1296
Vertex Figure Of :
{2,3,3} of size 48
{3,3,3} of size 120
{4,3,3} of size 192
{6,3,3} of size 240
{4,3,3} of size 384
{6,3,3} of size 720
{8,3,3} of size 768
{8,3,3} of size 768
{6,3,3} of size 1296
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6}*48, {6,3}*48
4-fold covers : {3,12}*96, {12,3}*96, {6,6}*96
6-fold covers : {3,6}*144, {6,3}*144
8-fold covers : {6,6}*192a, {3,6}*192, {6,3}*192, {6,12}*192a, {12,6}*192a, {6,12}*192b, {12,6}*192b, {6,6}*192b
10-fold covers : {6,15}*240, {15,6}*240
12-fold covers : {3,12}*288, {12,3}*288, {6,6}*288a, {6,6}*288b
14-fold covers : {6,21}*336, {21,6}*336
16-fold covers : {3,12}*384, {12,3}*384, {6,6}*384a, {6,6}*384b, {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
18-fold covers : {6,9}*432, {9,6}*432, {3,6}*432, {6,3}*432
20-fold covers : {12,15}*480, {15,12}*480, {6,30}*480, {30,6}*480
22-fold covers : {6,33}*528, {33,6}*528
24-fold covers : {3,6}*576, {6,3}*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, {12,3}*576
26-fold covers : {6,39}*624, {39,6}*624
27-fold covers : {3,9}*648, {9,3}*648, {9,9}*648a, {9,9}*648b
28-fold covers : {12,21}*672, {21,12}*672, {6,42}*672, {42,6}*672
30-fold covers : {6,15}*720e, {15,6}*720e
32-fold covers : {3,24}*768, {24,3}*768, {3,6}*768, {6,3}*768, {6,6}*768a, {6,12}*768a, {6,12}*768b, {12,6}*768a, {12,6}*768b, {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
34-fold covers : {6,51}*816, {51,6}*816
36-fold covers : {9,12}*864, {12,9}*864, {3,12}*864, {12,3}*864, {6,18}*864, {18,6}*864, {6,6}*864a, {6,6}*864b, {12,12}*864m, {6,6}*864c
38-fold covers : {6,57}*912, {57,6}*912
40-fold covers : {6,15}*960, {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
42-fold covers : {6,21}*1008b, {21,6}*1008b
44-fold covers : {12,33}*1056, {33,12}*1056, {6,66}*1056, {66,6}*1056
46-fold covers : {6,69}*1104, {69,6}*1104
48-fold covers : {3,12}*1152a, {12,3}*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, {12,3}*1152b, {12,6}*1152g, {24,3}*1152b, {3,24}*1152c, {24,3}*1152c, {6,12}*1152j, {12,6}*1152j
50-fold covers : {6,75}*1200, {75,6}*1200, {15,30}*1200, {30,15}*1200, {3,6}*1200, {6,3}*1200
52-fold covers : {12,39}*1248, {39,12}*1248, {6,78}*1248, {78,6}*1248
54-fold covers : {6,27}*1296, {27,6}*1296, {9,18}*1296a, {18,9}*1296a, {6,9}*1296a, {9,6}*1296a, {3,6}*1296, {6,3}*1296, {6,9}*1296b, {9,6}*1296b, {3,18}*1296a, {18,3}*1296a, {6,9}*1296c, {9,6}*1296c, {6,9}*1296d, {9,6}*1296d, {6,6}*1296a, {6,6}*1296b, {6,9}*1296e, {9,6}*1296e, {3,18}*1296b, {6,9}*1296f, {9,6}*1296f, {9,18}*1296b, {9,18}*1296c, {18,3}*1296b, {18,9}*1296b, {18,9}*1296c
56-fold covers : {6,21}*1344, {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
58-fold covers : {6,87}*1392, {87,6}*1392
60-fold covers : {12,15}*1440c, {15,12}*1440c, {3,15}*1440, {15,3}*1440, {15,15}*1440, {6,30}*1440g, {30,6}*1440g, {6,30}*1440h, {30,6}*1440h
62-fold covers : {6,93}*1488, {93,6}*1488
66-fold covers : {6,33}*1584, {33,6}*1584
68-fold covers : {12,51}*1632, {51,12}*1632, {6,102}*1632, {102,6}*1632
70-fold covers : {6,105}*1680, {105,6}*1680
72-fold covers : {6,9}*1728, {9,6}*1728, {3,6}*1728, {6,3}*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, {12,9}*1728, {3,12}*1728, {12,3}*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
74-fold covers : {6,111}*1776, {111,6}*1776
76-fold covers : {12,57}*1824, {57,12}*1824, {6,114}*1824, {114,6}*1824
78-fold covers : {6,39}*1872, {39,6}*1872
80-fold covers : {12,15}*1920, {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, {15,15}*1920
81-fold covers : {9,9}*1944a, {9,9}*1944b, {9,9}*1944c, {9,9}*1944d
82-fold covers : {6,123}*1968, {123,6}*1968
Permutation Representation (GAP) :
```s0 := (3,4);;
s1 := (2,3);;
s2 := (1,2);;
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(4)!(3,4);
s1 := Sym(4)!(2,3);
s2 := Sym(4)!(1,2);
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 >;

```
References : None.
to this polytope