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# Polytope of Type {4,6}

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

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

```
References : None.
to this polytope