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# Polytope of Type {54}

Atlas Canonical Name : {54}*108
Also Known As : 54-gon, {54}. if this polytope has another name.
Group : SmallGroup(108,4)
Rank : 2
Schlafli Type : {54}
Number of vertices, edges, etc : 54, 54
Order of s0s1 : 54
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{54,2} of size 216
{54,4} of size 432
{54,4} of size 432
{54,4} of size 432
{54,6} of size 648
{54,6} of size 648
{54,8} of size 864
{54,4} of size 864
{54,6} of size 972
{54,6} of size 972
{54,6} of size 972
{54,10} of size 1080
{54,12} of size 1296
{54,12} of size 1296
{54,12} of size 1296
{54,14} of size 1512
{54,16} of size 1728
{54,4} of size 1728
{54,8} of size 1728
{54,4} of size 1728
{54,8} of size 1728
{54,8} of size 1728
{54,18} of size 1944
{54,18} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
{54,6} of size 1944
Vertex Figure Of :
{2,54} of size 216
{4,54} of size 432
{4,54} of size 432
{4,54} of size 432
{6,54} of size 648
{6,54} of size 648
{8,54} of size 864
{4,54} of size 864
{6,54} of size 972
{6,54} of size 972
{6,54} of size 972
{10,54} of size 1080
{12,54} of size 1296
{12,54} of size 1296
{12,54} of size 1296
{14,54} of size 1512
{16,54} of size 1728
{4,54} of size 1728
{8,54} of size 1728
{4,54} of size 1728
{8,54} of size 1728
{8,54} of size 1728
{18,54} of size 1944
{18,54} of size 1944
{6,54} of size 1944
{6,54} of size 1944
{6,54} of size 1944
{6,54} of size 1944
{6,54} of size 1944
{6,54} of size 1944
{6,54} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {27}*54
3-fold quotients : {18}*36
6-fold quotients : {9}*18
9-fold quotients : {6}*12
18-fold quotients : {3}*6
27-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {108}*216
3-fold covers : {162}*324
4-fold covers : {216}*432
5-fold covers : {270}*540
6-fold covers : {324}*648
7-fold covers : {378}*756
8-fold covers : {432}*864
9-fold covers : {486}*972
10-fold covers : {540}*1080
11-fold covers : {594}*1188
12-fold covers : {648}*1296
13-fold covers : {702}*1404
14-fold covers : {756}*1512
15-fold covers : {810}*1620
16-fold covers : {864}*1728
17-fold covers : {918}*1836
18-fold covers : {972}*1944
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)
(45,46)(47,48)(49,50)(51,52)(53,54);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,21)(18,19)
(20,25)(22,23)(24,29)(26,27)(28,33)(30,31)(32,37)(34,35)(36,41)(38,39)(40,45)
(42,43)(44,49)(46,47)(48,53)(50,51)(52,54);;
poly := Group([s0,s1]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(54)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46)(47,48)(49,50)(51,52)(53,54);
s1 := Sym(54)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,21)
(18,19)(20,25)(22,23)(24,29)(26,27)(28,33)(30,31)(32,37)(34,35)(36,41)(38,39)
(40,45)(42,43)(44,49)(46,47)(48,53)(50,51)(52,54);
poly := sub<Sym(54)|s0,s1>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope