Polytope of Type {6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3}*108
Also Known As : {6,3}(3,0), {6,3}6. if this polytope has another name.
Group : SmallGroup(108,17)
Rank : 3
Schlafli Type : {6,3}
Number of vertices, edges, etc : 18, 27, 9
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,3,2} of size 216
{6,3,4} of size 432
{6,3,6} of size 648
{6,3,4} of size 864
{6,3,3} of size 1296
{6,3,4} of size 1296
{6,3,8} of size 1728
{6,3,6} of size 1944
{6,3,6} of size 1944
{6,3,6} of size 1944
Vertex Figure Of :
{2,6,3} of size 216
{3,6,3} of size 324
{4,6,3} of size 432
{6,6,3} of size 648
{6,6,3} of size 648
{8,6,3} of size 864
{3,6,3} of size 972
{9,6,3} of size 972
{10,6,3} of size 1080
{12,6,3} of size 1296
{12,6,3} of size 1296
{14,6,3} of size 1512
{15,6,3} of size 1620
{16,6,3} of size 1728
{4,6,3} of size 1728
{6,6,3} of size 1944
{18,6,3} of size 1944
{18,6,3} of size 1944
{6,6,3} of size 1944
{6,6,3} of size 1944
{6,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,3}*36
9-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6}*216a
3-fold covers : {6,9}*324a, {6,9}*324b, {6,9}*324c, {6,9}*324d, {6,3}*324, {18,3}*324
4-fold covers : {6,12}*432a, {12,6}*432c, {6,3}*432, {12,3}*432
5-fold covers : {6,15}*540
6-fold covers : {6,18}*648a, {6,18}*648c, {6,18}*648d, {6,18}*648e, {6,6}*648c, {18,6}*648h, {6,6}*648f
7-fold covers : {6,21}*756
8-fold covers : {6,24}*864a, {12,12}*864b, {24,6}*864c, {12,3}*864, {24,3}*864, {6,6}*864b, {12,6}*864a
9-fold covers : {18,9}*972a, {18,3}*972a, {6,9}*972a, {6,9}*972b, {18,9}*972b, {6,9}*972c, {18,9}*972c, {18,9}*972d, {18,9}*972e, {6,27}*972a, {6,9}*972d, {18,9}*972f, {18,9}*972g, {18,9}*972h, {18,9}*972i, {6,9}*972e, {18,9}*972j, {6,27}*972b, {6,27}*972c, {6,3}*972, {18,3}*972b
10-fold covers : {30,6}*1080a, {6,30}*1080b
11-fold covers : {6,33}*1188
12-fold covers : {6,36}*1296a, {6,36}*1296c, {6,36}*1296d, {6,36}*1296e, {18,12}*1296d, {6,12}*1296c, {12,18}*1296e, {12,18}*1296f, {12,18}*1296g, {12,18}*1296h, {12,6}*1296d, {36,6}*1296h, {6,9}*1296a, {6,3}*1296, {36,3}*1296, {6,9}*1296b, {12,3}*1296a, {18,3}*1296a, {12,9}*1296a, {6,9}*1296c, {12,9}*1296b, {12,9}*1296c, {6,9}*1296d, {12,9}*1296d, {6,12}*1296h, {12,6}*1296i
13-fold covers : {6,39}*1404
14-fold covers : {42,6}*1512a, {6,42}*1512b
15-fold covers : {6,45}*1620a, {6,45}*1620b, {6,45}*1620c, {6,45}*1620d, {6,15}*1620, {18,15}*1620
16-fold covers : {6,48}*1728a, {24,12}*1728a, {12,12}*1728b, {24,12}*1728b, {12,24}*1728c, {12,24}*1728e, {48,6}*1728c, {6,3}*1728, {24,3}*1728, {12,12}*1728k, {6,12}*1728a, {12,12}*1728n, {12,6}*1728c, {24,6}*1728b, {6,6}*1728a, {24,6}*1728d, {6,12}*1728d, {12,6}*1728e, {12,6}*1728f, {12,3}*1728, {6,6}*1728e
17-fold covers : {6,51}*1836
18-fold covers : {18,18}*1944b, {6,18}*1944a, {18,6}*1944b, {6,18}*1944d, {18,18}*1944e, {6,18}*1944f, {18,18}*1944g, {18,18}*1944j, {18,18}*1944n, {6,54}*1944a, {6,18}*1944h, {18,18}*1944p, {18,18}*1944r, {18,18}*1944w, {18,18}*1944aa, {6,18}*1944i, {18,18}*1944ac, {6,54}*1944c, {6,54}*1944e, {6,6}*1944c, {18,6}*1944k, {6,18}*1944m, {18,6}*1944o, {6,6}*1944d, {6,6}*1944f, {6,18}*1944p, {6,18}*1944q, {6,18}*1944r, {6,6}*1944i, {18,6}*1944u
Permutation Representation (GAP) :
s0 := (4,5)(6,7)(8,9);;
s1 := (2,6)(3,4)(5,7);;
s2 := (1,2)(4,8)(5,9);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(9)!(4,5)(6,7)(8,9);
s1 := Sym(9)!(2,6)(3,4)(5,7);
s2 := Sym(9)!(1,2)(4,8)(5,9);
poly := sub<Sym(9)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >;
References : None.
to this polytope