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# Polytope of Type {6,4}

Atlas Canonical Name : {6,4}*72
if this polytope has a name.
Group : SmallGroup(72,40)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 9, 18, 6
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{6,4,2} of size 144
{6,4,4} of size 288
{6,4,6} of size 432
{6,4,8} of size 576
{6,4,10} of size 720
{6,4,12} of size 864
{6,4,14} of size 1008
{6,4,16} of size 1152
{6,4,4} of size 1152
{6,4,4} of size 1152
{6,4,18} of size 1296
{6,4,20} of size 1440
{6,4,22} of size 1584
{6,4,24} of size 1728
{6,4,26} of size 1872
Vertex Figure Of :
{2,6,4} of size 144
{4,6,4} of size 720
{3,6,4} of size 1152
{4,6,4} of size 1152
{4,6,4} of size 1152
{6,6,4} of size 1296
{4,6,4} of size 1440
{4,6,4} of size 1440
{4,6,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,4}*144
3-fold covers : {6,4}*216, {6,12}*216a, {6,12}*216b, {6,12}*216c
4-fold covers : {6,8}*288, {12,4}*288
5-fold covers : {6,20}*360
6-fold covers : {6,4}*432a, {6,12}*432e, {6,12}*432f, {6,4}*432b, {6,12}*432h, {6,12}*432i
7-fold covers : {6,28}*504
8-fold covers : {6,16}*576, {12,4}*576, {12,8}*576a, {24,4}*576a, {24,4}*576b, {12,8}*576b
9-fold covers : {18,4}*648, {6,36}*648a, {6,12}*648, {6,36}*648b, {6,36}*648c
10-fold covers : {30,4}*720, {6,20}*720
11-fold covers : {6,44}*792
12-fold covers : {6,8}*864a, {6,24}*864d, {6,24}*864e, {12,4}*864b, {12,12}*864f, {12,12}*864g, {12,4}*864d, {12,12}*864j, {6,8}*864b, {6,24}*864g, {6,24}*864h, {12,12}*864l, {12,12}*864o
13-fold covers : {6,52}*936
14-fold covers : {42,4}*1008, {6,28}*1008
15-fold covers : {6,20}*1080, {6,60}*1080a, {6,60}*1080b, {6,60}*1080c
16-fold covers : {24,4}*1152a, {12,8}*1152a, {24,8}*1152a, {24,8}*1152b, {24,8}*1152c, {24,8}*1152d, {48,4}*1152a, {12,16}*1152a, {48,4}*1152b, {12,16}*1152b, {12,4}*1152a, {12,8}*1152b, {24,4}*1152b, {6,32}*1152, {6,4}*1152, {6,8}*1152a, {6,8}*1152b, {6,8}*1152c, {12,4}*1152b, {12,8}*1152c
17-fold covers : {6,68}*1224
18-fold covers : {18,4}*1296a, {18,4}*1296b, {6,4}*1296a, {6,12}*1296j, {6,12}*1296k, {6,12}*1296l, {6,12}*1296m, {6,12}*1296n, {6,36}*1296m, {6,12}*1296o, {6,36}*1296n, {6,36}*1296o, {6,12}*1296s, {6,12}*1296t, {6,12}*1296u
19-fold covers : {6,76}*1368
20-fold covers : {60,4}*1440, {30,8}*1440, {6,40}*1440, {12,20}*1440
21-fold covers : {6,28}*1512, {6,84}*1512a, {6,84}*1512b, {6,84}*1512c
22-fold covers : {66,4}*1584, {6,44}*1584
23-fold covers : {6,92}*1656
24-fold covers : {6,16}*1728a, {6,48}*1728d, {6,48}*1728e, {12,4}*1728b, {12,12}*1728f, {12,12}*1728g, {12,8}*1728a, {12,24}*1728g, {12,24}*1728h, {24,4}*1728a, {24,12}*1728i, {24,12}*1728j, {24,4}*1728c, {24,12}*1728k, {24,12}*1728l, {12,8}*1728d, {12,24}*1728m, {12,24}*1728n, {24,4}*1728f, {24,12}*1728q, {24,4}*1728g, {24,12}*1728r, {6,16}*1728b, {6,48}*1728g, {12,8}*1728g, {12,24}*1728s, {12,8}*1728h, {12,24}*1728t, {12,4}*1728c, {12,12}*1728q, {6,48}*1728h, {12,12}*1728t, {12,24}*1728u, {24,12}*1728v, {24,12}*1728w, {12,24}*1728x, {6,4}*1728, {6,12}*1728j, {12,12}*1728ab
25-fold covers : {6,100}*1800, {30,4}*1800
26-fold covers : {78,4}*1872, {6,52}*1872
27-fold covers : {18,4}*1944a, {18,12}*1944a, {18,12}*1944b, {6,4}*1944, {6,12}*1944a, {6,12}*1944b, {18,4}*1944b, {18,12}*1944c, {18,12}*1944d, {18,4}*1944c, {18,12}*1944e, {18,12}*1944f, {18,12}*1944g, {6,36}*1944, {6,12}*1944c, {18,12}*1944h, {18,12}*1944i, {6,108}*1944a, {6,108}*1944b, {6,108}*1944c
Permutation Representation (GAP) :
```s0 := (3,4)(5,6);;
s1 := (2,3);;
s2 := (1,2)(3,5)(4,6);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(6)!(3,4)(5,6);
s1 := Sym(6)!(2,3);
s2 := Sym(6)!(1,2)(3,5)(4,6);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope