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# Polytope of Type {3,2,3}

Atlas Canonical Name : {3,2,3}*36
if this polytope has a name.
Group : SmallGroup(36,10)
Rank : 4
Schlafli Type : {3,2,3}
Number of vertices, edges, etc : 3, 3, 3, 3
Order of s0s1s2s3 : 3
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Projective
Locally Projective
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,2,3,2} of size 72
{3,2,3,3} of size 144
{3,2,3,4} of size 144
{3,2,3,6} of size 216
{3,2,3,4} of size 288
{3,2,3,6} of size 288
{3,2,3,5} of size 360
{3,2,3,8} of size 576
{3,2,3,12} of size 576
{3,2,3,6} of size 648
{3,2,3,5} of size 720
{3,2,3,10} of size 720
{3,2,3,10} of size 720
{3,2,3,6} of size 864
{3,2,3,12} of size 864
{3,2,3,6} of size 1152
{3,2,3,8} of size 1152
{3,2,3,10} of size 1440
{3,2,3,12} of size 1728
{3,2,3,24} of size 1728
{3,2,3,6} of size 1800
{3,2,3,10} of size 1800
{3,2,3,6} of size 1944
{3,2,3,18} of size 1944
Vertex Figure Of :
{2,3,2,3} of size 72
{3,3,2,3} of size 144
{4,3,2,3} of size 144
{6,3,2,3} of size 216
{4,3,2,3} of size 288
{6,3,2,3} of size 288
{5,3,2,3} of size 360
{8,3,2,3} of size 576
{12,3,2,3} of size 576
{6,3,2,3} of size 648
{5,3,2,3} of size 720
{10,3,2,3} of size 720
{10,3,2,3} of size 720
{6,3,2,3} of size 864
{12,3,2,3} of size 864
{6,3,2,3} of size 1152
{8,3,2,3} of size 1152
{10,3,2,3} of size 1440
{12,3,2,3} of size 1728
{24,3,2,3} of size 1728
{6,3,2,3} of size 1800
{10,3,2,3} of size 1800
{6,3,2,3} of size 1944
{18,3,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,2,6}*72, {6,2,3}*72
3-fold covers : {3,2,9}*108, {9,2,3}*108, {3,6,3}*108
4-fold covers : {3,2,12}*144, {12,2,3}*144, {6,2,6}*144
5-fold covers : {3,2,15}*180, {15,2,3}*180
6-fold covers : {3,2,18}*216, {6,2,9}*216, {9,2,6}*216, {18,2,3}*216, {3,6,6}*216a, {6,6,3}*216a, {3,6,6}*216b, {6,6,3}*216b
7-fold covers : {3,2,21}*252, {21,2,3}*252
8-fold covers : {3,2,24}*288, {24,2,3}*288, {6,2,12}*288, {12,2,6}*288, {6,4,6}*288, {3,4,6}*288, {6,4,3}*288
9-fold covers : {9,2,9}*324, {3,6,9}*324, {9,6,3}*324, {3,2,27}*324, {27,2,3}*324, {3,6,3}*324a, {3,6,3}*324b
10-fold covers : {3,2,30}*360, {6,2,15}*360, {15,2,6}*360, {30,2,3}*360
11-fold covers : {3,2,33}*396, {33,2,3}*396
12-fold covers : {3,2,36}*432, {36,2,3}*432, {9,2,12}*432, {12,2,9}*432, {3,6,12}*432a, {12,6,3}*432a, {6,2,18}*432, {18,2,6}*432, {6,6,6}*432a, {3,6,12}*432b, {12,6,3}*432b, {6,6,6}*432b, {6,6,6}*432c, {6,6,6}*432g
13-fold covers : {3,2,39}*468, {39,2,3}*468
14-fold covers : {3,2,42}*504, {6,2,21}*504, {21,2,6}*504, {42,2,3}*504
15-fold covers : {3,2,45}*540, {45,2,3}*540, {9,2,15}*540, {15,2,9}*540, {3,6,15}*540, {15,6,3}*540
16-fold covers : {3,2,48}*576, {48,2,3}*576, {12,2,12}*576, {6,4,12}*576, {12,4,6}*576, {6,2,24}*576, {24,2,6}*576, {6,8,6}*576, {3,4,12}*576, {12,4,3}*576, {3,8,6}*576, {6,8,3}*576, {3,4,3}*576, {6,4,6}*576a, {6,4,6}*576b
17-fold covers : {3,2,51}*612, {51,2,3}*612
18-fold covers : {9,2,18}*648, {18,2,9}*648, {3,6,18}*648a, {6,6,9}*648a, {9,6,6}*648a, {18,6,3}*648a, {3,2,54}*648, {6,2,27}*648, {27,2,6}*648, {54,2,3}*648, {3,6,6}*648a, {3,6,6}*648b, {6,6,3}*648a, {6,6,3}*648b, {3,6,18}*648b, {6,6,9}*648b, {9,6,6}*648b, {18,6,3}*648b, {3,6,6}*648c, {3,6,6}*648d, {3,6,6}*648e, {6,6,3}*648c, {6,6,3}*648d, {6,6,3}*648e
19-fold covers : {3,2,57}*684, {57,2,3}*684
20-fold covers : {12,2,15}*720, {15,2,12}*720, {3,2,60}*720, {60,2,3}*720, {6,10,6}*720, {6,2,30}*720, {30,2,6}*720
21-fold covers : {3,2,63}*756, {63,2,3}*756, {9,2,21}*756, {21,2,9}*756, {3,6,21}*756, {21,6,3}*756
22-fold covers : {3,2,66}*792, {6,2,33}*792, {33,2,6}*792, {66,2,3}*792
23-fold covers : {3,2,69}*828, {69,2,3}*828
24-fold covers : {3,2,72}*864, {72,2,3}*864, {9,2,24}*864, {24,2,9}*864, {3,6,24}*864a, {24,6,3}*864a, {6,2,36}*864, {36,2,6}*864, {12,2,18}*864, {18,2,12}*864, {6,6,12}*864a, {12,6,6}*864a, {6,4,18}*864, {18,4,6}*864, {6,12,6}*864a, {3,6,24}*864b, {24,6,3}*864b, {3,4,18}*864, {18,4,3}*864, {6,4,9}*864, {9,4,6}*864, {3,12,6}*864a, {6,12,3}*864a, {6,6,12}*864b, {6,6,12}*864c, {6,12,6}*864b, {12,6,6}*864b, {12,6,6}*864d, {6,6,12}*864e, {12,6,6}*864e, {6,12,6}*864f, {6,12,6}*864g, {3,6,6}*864, {3,12,6}*864b, {6,6,3}*864, {6,12,3}*864b
25-fold covers : {3,2,75}*900, {75,2,3}*900, {15,2,15}*900
26-fold covers : {3,2,78}*936, {6,2,39}*936, {39,2,6}*936, {78,2,3}*936
27-fold covers : {9,6,9}*972, {3,6,3}*972, {9,2,27}*972, {27,2,9}*972, {3,6,27}*972, {27,6,3}*972, {3,6,9}*972a, {9,6,3}*972a, {3,6,9}*972b, {9,6,3}*972b, {3,2,81}*972, {81,2,3}*972
28-fold covers : {12,2,21}*1008, {21,2,12}*1008, {3,2,84}*1008, {84,2,3}*1008, {6,14,6}*1008, {6,2,42}*1008, {42,2,6}*1008
29-fold covers : {3,2,87}*1044, {87,2,3}*1044
30-fold covers : {3,2,90}*1080, {6,2,45}*1080, {45,2,6}*1080, {90,2,3}*1080, {9,2,30}*1080, {15,2,18}*1080, {18,2,15}*1080, {30,2,9}*1080, {3,6,30}*1080a, {6,6,15}*1080a, {15,6,6}*1080a, {30,6,3}*1080a, {3,6,30}*1080b, {6,6,15}*1080b, {15,6,6}*1080b, {30,6,3}*1080b
31-fold covers : {3,2,93}*1116, {93,2,3}*1116
32-fold covers : {3,2,96}*1152, {96,2,3}*1152, {12,4,12}*1152, {6,8,12}*1152a, {12,8,6}*1152a, {6,4,24}*1152a, {24,4,6}*1152a, {6,8,12}*1152b, {12,8,6}*1152b, {6,4,24}*1152b, {24,4,6}*1152b, {6,4,12}*1152a, {12,4,6}*1152a, {12,2,24}*1152, {24,2,12}*1152, {6,16,6}*1152, {6,2,48}*1152, {48,2,6}*1152, {3,8,12}*1152, {12,8,3}*1152, {3,4,12}*1152, {12,4,3}*1152, {3,8,6}*1152, {6,8,3}*1152, {3,4,24}*1152, {24,4,3}*1152, {3,4,3}*1152, {3,4,6}*1152a, {6,4,3}*1152a, {6,4,12}*1152b, {12,4,6}*1152b, {6,4,12}*1152c, {12,4,6}*1152c, {6,4,6}*1152a, {6,4,6}*1152b, {6,4,12}*1152d, {12,4,6}*1152d, {6,8,6}*1152a, {6,8,6}*1152b, {6,8,6}*1152c, {6,8,6}*1152d, {3,4,6}*1152b, {6,4,3}*1152b
33-fold covers : {3,2,99}*1188, {99,2,3}*1188, {9,2,33}*1188, {33,2,9}*1188, {3,6,33}*1188, {33,6,3}*1188
34-fold covers : {3,2,102}*1224, {6,2,51}*1224, {51,2,6}*1224, {102,2,3}*1224
35-fold covers : {15,2,21}*1260, {21,2,15}*1260, {3,2,105}*1260, {105,2,3}*1260
36-fold covers : {9,2,36}*1296, {36,2,9}*1296, {9,6,12}*1296a, {12,6,9}*1296a, {3,6,36}*1296a, {36,6,3}*1296a, {12,2,27}*1296, {27,2,12}*1296, {3,2,108}*1296, {108,2,3}*1296, {3,6,12}*1296a, {12,6,3}*1296a, {3,6,12}*1296b, {12,6,3}*1296b, {18,2,18}*1296, {6,6,18}*1296a, {18,6,6}*1296a, {6,2,54}*1296, {54,2,6}*1296, {6,6,6}*1296a, {6,6,6}*1296b, {3,6,36}*1296b, {36,6,3}*1296b, {9,6,12}*1296b, {12,6,9}*1296b, {3,6,12}*1296c, {3,6,12}*1296d, {12,6,3}*1296c, {12,6,3}*1296d, {3,6,12}*1296e, {12,6,3}*1296e, {6,6,18}*1296b, {6,6,18}*1296c, {6,6,18}*1296e, {6,18,6}*1296a, {18,6,6}*1296b, {18,6,6}*1296c, {18,6,6}*1296e, {6,6,6}*1296c, {6,6,6}*1296f, {6,6,6}*1296g, {6,6,6}*1296j, {6,6,6}*1296k, {6,6,6}*1296n, {6,6,6}*1296o, {6,6,6}*1296p, {3,6,12}*1296f, {12,6,3}*1296f, {6,6,6}*1296q, {6,6,6}*1296s
37-fold covers : {3,2,111}*1332, {111,2,3}*1332
38-fold covers : {3,2,114}*1368, {6,2,57}*1368, {57,2,6}*1368, {114,2,3}*1368
39-fold covers : {3,2,117}*1404, {117,2,3}*1404, {9,2,39}*1404, {39,2,9}*1404, {3,6,39}*1404, {39,6,3}*1404
40-fold covers : {15,2,24}*1440, {24,2,15}*1440, {3,2,120}*1440, {120,2,3}*1440, {6,10,12}*1440, {12,10,6}*1440, {6,20,6}*1440, {12,2,30}*1440, {30,2,12}*1440, {6,2,60}*1440, {60,2,6}*1440, {6,4,30}*1440, {30,4,6}*1440, {6,4,15}*1440, {15,4,6}*1440, {3,4,30}*1440, {30,4,3}*1440
41-fold covers : {3,2,123}*1476, {123,2,3}*1476
42-fold covers : {3,2,126}*1512, {6,2,63}*1512, {63,2,6}*1512, {126,2,3}*1512, {9,2,42}*1512, {18,2,21}*1512, {21,2,18}*1512, {42,2,9}*1512, {3,6,42}*1512a, {6,6,21}*1512a, {21,6,6}*1512a, {42,6,3}*1512a, {3,6,42}*1512b, {6,6,21}*1512b, {21,6,6}*1512b, {42,6,3}*1512b
43-fold covers : {3,2,129}*1548, {129,2,3}*1548
44-fold covers : {12,2,33}*1584, {33,2,12}*1584, {3,2,132}*1584, {132,2,3}*1584, {6,22,6}*1584, {6,2,66}*1584, {66,2,6}*1584
45-fold covers : {9,2,45}*1620, {45,2,9}*1620, {3,6,45}*1620, {45,6,3}*1620, {9,6,15}*1620, {15,6,9}*1620, {3,2,135}*1620, {135,2,3}*1620, {15,2,27}*1620, {27,2,15}*1620, {3,6,15}*1620a, {15,6,3}*1620a, {3,6,15}*1620b, {15,6,3}*1620b
46-fold covers : {3,2,138}*1656, {6,2,69}*1656, {69,2,6}*1656, {138,2,3}*1656
47-fold covers : {3,2,141}*1692, {141,2,3}*1692
48-fold covers : {3,2,144}*1728, {144,2,3}*1728, {9,2,48}*1728, {48,2,9}*1728, {3,6,48}*1728a, {48,6,3}*1728a, {12,2,36}*1728, {36,2,12}*1728, {12,6,12}*1728a, {12,4,18}*1728, {18,4,12}*1728, {6,4,36}*1728, {36,4,6}*1728, {6,12,12}*1728a, {12,12,6}*1728a, {6,2,72}*1728, {72,2,6}*1728, {18,2,24}*1728, {24,2,18}*1728, {6,6,24}*1728a, {24,6,6}*1728a, {6,8,18}*1728, {18,8,6}*1728, {6,24,6}*1728a, {3,6,48}*1728b, {48,6,3}*1728b, {3,4,36}*1728, {36,4,3}*1728, {3,8,18}*1728, {18,8,3}*1728, {9,4,12}*1728, {12,4,9}*1728, {3,12,12}*1728a, {12,12,3}*1728a, {6,8,9}*1728, {9,8,6}*1728, {3,24,6}*1728a, {6,24,3}*1728a, {6,6,24}*1728b, {6,6,24}*1728c, {6,24,6}*1728b, {24,6,6}*1728b, {24,6,6}*1728d, {6,6,24}*1728e, {24,6,6}*1728e, {12,6,12}*1728b, {12,6,12}*1728e, {12,6,12}*1728f, {6,12,12}*1728b, {6,12,12}*1728c, {12,12,6}*1728b, {12,12,6}*1728f, {6,24,6}*1728f, {6,24,6}*1728g, {6,12,12}*1728g, {12,12,6}*1728g, {3,4,9}*1728, {9,4,3}*1728, {3,12,3}*1728, {6,4,18}*1728a, {18,4,6}*1728a, {6,4,18}*1728b, {18,4,6}*1728b, {6,12,6}*1728a, {6,12,6}*1728b, {3,12,6}*1728, {3,24,6}*1728b, {6,12,3}*1728, {6,24,3}*1728b, {3,6,12}*1728, {3,12,12}*1728b, {12,6,3}*1728, {12,12,3}*1728b, {6,6,6}*1728a, {6,6,6}*1728f, {6,6,12}*1728a, {6,12,6}*1728e, {6,12,6}*1728f, {6,12,6}*1728h, {6,12,6}*1728i, {6,12,6}*1728j, {6,12,6}*1728l, {12,6,6}*1728a
49-fold covers : {3,2,147}*1764, {147,2,3}*1764, {21,2,21}*1764
50-fold covers : {3,2,150}*1800, {6,2,75}*1800, {75,2,6}*1800, {150,2,3}*1800, {3,10,6}*1800, {6,10,3}*1800, {6,10,15}*1800, {15,10,6}*1800, {15,2,30}*1800, {30,2,15}*1800
51-fold covers : {3,2,153}*1836, {153,2,3}*1836, {9,2,51}*1836, {51,2,9}*1836, {3,6,51}*1836, {51,6,3}*1836
52-fold covers : {12,2,39}*1872, {39,2,12}*1872, {3,2,156}*1872, {156,2,3}*1872, {6,26,6}*1872, {6,2,78}*1872, {78,2,6}*1872
53-fold covers : {3,2,159}*1908, {159,2,3}*1908
54-fold covers : {9,6,18}*1944a, {18,6,9}*1944a, {3,6,6}*1944a, {6,6,3}*1944a, {9,2,54}*1944, {18,2,27}*1944, {27,2,18}*1944, {54,2,9}*1944, {3,6,54}*1944a, {6,6,27}*1944a, {27,6,6}*1944a, {54,6,3}*1944a, {3,6,18}*1944a, {6,6,9}*1944a, {9,6,6}*1944a, {18,6,3}*1944a, {3,6,18}*1944b, {6,6,9}*1944b, {9,6,6}*1944b, {18,6,3}*1944b, {3,2,162}*1944, {6,2,81}*1944, {81,2,6}*1944, {162,2,3}*1944, {6,18,9}*1944, {9,6,18}*1944b, {9,18,6}*1944, {18,6,9}*1944b, {3,6,18}*1944c, {3,6,18}*1944d, {6,6,9}*1944c, {6,6,9}*1944d, {9,6,6}*1944c, {9,6,6}*1944d, {18,6,3}*1944c, {18,6,3}*1944d, {3,6,18}*1944e, {6,6,9}*1944e, {9,6,6}*1944e, {18,6,3}*1944e, {3,6,6}*1944b, {3,6,6}*1944c, {3,6,6}*1944d, {6,6,3}*1944b, {6,6,3}*1944c, {6,6,3}*1944d, {3,6,54}*1944b, {6,6,27}*1944b, {27,6,6}*1944b, {54,6,3}*1944b, {3,6,6}*1944e, {3,6,6}*1944f, {3,6,6}*1944g, {6,6,3}*1944e, {6,6,3}*1944f, {6,6,3}*1944g, {6,6,9}*1944f, {6,6,9}*1944g, {9,6,6}*1944f, {9,6,6}*1944g, {6,6,9}*1944h, {9,6,6}*1944h, {3,6,6}*1944h, {3,18,6}*1944, {6,6,3}*1944h, {6,18,3}*1944
55-fold covers : {15,2,33}*1980, {33,2,15}*1980, {3,2,165}*1980, {165,2,3}*1980
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2);;
s2 := (5,6);;
s3 := (4,5);;
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(6)!(2,3);
s1 := Sym(6)!(1,2);
s2 := Sym(6)!(5,6);
s3 := Sym(6)!(4,5);
poly := sub<Sym(6)|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 >;

```

to this polytope