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# Polytope of Type {18,6}

Atlas Canonical Name : {18,6}*216b
if this polytope has a name.
Group : SmallGroup(216,101)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 18, 54, 6
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,6,2} of size 432
{18,6,3} of size 648
{18,6,4} of size 864
{18,6,6} of size 1296
{18,6,6} of size 1296
{18,6,8} of size 1728
{18,6,9} of size 1944
{18,6,3} of size 1944
Vertex Figure Of :
{2,18,6} of size 432
{4,18,6} of size 864
{4,18,6} of size 864
{4,18,6} of size 864
{6,18,6} of size 1296
{6,18,6} of size 1296
{8,18,6} of size 1728
{4,18,6} of size 1728
{6,18,6} of size 1944
{6,18,6} of size 1944
{6,18,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,6}*108
3-fold quotients : {18,2}*72, {6,6}*72c
6-fold quotients : {9,2}*36, {3,6}*36
9-fold quotients : {6,2}*24
18-fold quotients : {3,2}*12
27-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,6}*432b, {18,12}*432b
3-fold covers : {18,18}*648c, {18,6}*648a, {54,6}*648b, {18,6}*648i
4-fold covers : {72,6}*864b, {36,12}*864b, {18,24}*864b, {18,6}*864, {18,12}*864b
5-fold covers : {18,30}*1080a, {90,6}*1080b
6-fold covers : {36,18}*1296b, {36,6}*1296a, {108,6}*1296b, {18,36}*1296c, {18,12}*1296e, {54,12}*1296b, {36,6}*1296l, {18,12}*1296l
7-fold covers : {18,42}*1512a, {126,6}*1512b
8-fold covers : {144,6}*1728b, {36,24}*1728a, {36,12}*1728b, {36,24}*1728b, {72,12}*1728b, {72,12}*1728d, {18,48}*1728b, {36,6}*1728a, {18,12}*1728a, {18,6}*1728a, {36,6}*1728c, {18,12}*1728b, {36,12}*1728f, {36,12}*1728g, {18,24}*1728b, {18,24}*1728d, {18,12}*1728d
9-fold covers : {18,18}*1944a, {54,18}*1944b, {54,6}*1944a, {18,6}*1944h, {18,18}*1944u, {18,18}*1944y, {18,6}*1944i, {54,6}*1944c, {54,6}*1944e, {162,6}*1944b, {18,18}*1944ad, {18,18}*1944af, {18,6}*1944m, {18,6}*1944n, {18,6}*1944o, {54,6}*1944g
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)
(16,23)(17,22)(18,24)(29,30)(31,34)(32,36)(33,35)(37,47)(38,46)(39,48)(40,53)
(41,52)(42,54)(43,50)(44,49)(45,51);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)(10,31)
(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,50)(20,49)(21,51)
(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*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)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,20)(11,19)(12,21)(13,26)(14,25)
(15,27)(16,23)(17,22)(18,24)(29,30)(31,34)(32,36)(33,35)(37,47)(38,46)(39,48)
(40,53)(41,52)(42,54)(43,50)(44,49)(45,51);
s1 := Sym(54)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)
(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,50)(20,49)
(21,51)(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);
s2 := Sym(54)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);
poly := sub<Sym(54)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*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