Polytope of Type {6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*108
if this polytope has a name.
Group : SmallGroup(108,16)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 6, 27, 9
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,9,2} of size 216
{6,9,4} of size 432
{6,9,6} of size 648
{6,9,4} of size 864
{6,9,8} of size 1728
{6,9,18} of size 1944
{6,9,6} of size 1944
{6,9,6} of size 1944
{6,9,6} of size 1944
{6,9,6} of size 1944
Vertex Figure Of :
{2,6,9} of size 216
{3,6,9} of size 324
{4,6,9} of size 432
{6,6,9} of size 648
{6,6,9} of size 648
{8,6,9} of size 864
{9,6,9} of size 972
{3,6,9} of size 972
{10,6,9} of size 1080
{12,6,9} of size 1296
{12,6,9} of size 1296
{4,6,9} of size 1296
{14,6,9} of size 1512
{15,6,9} of size 1620
{16,6,9} of size 1728
{4,6,9} of size 1728
{18,6,9} of size 1944
{6,6,9} of size 1944
{18,6,9} of size 1944
{6,6,9} of size 1944
{6,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,9}*36, {6,3}*36
9-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,18}*216b
3-fold covers : {18,9}*324, {6,9}*324a, {6,27}*324
4-fold covers : {6,36}*432b, {12,18}*432b, {6,9}*432, {12,9}*432
5-fold covers : {6,45}*540
6-fold covers : {18,18}*648b, {6,18}*648a, {6,54}*648b, {6,18}*648i
7-fold covers : {6,63}*756
8-fold covers : {6,72}*864b, {12,36}*864b, {24,18}*864b, {12,9}*864, {24,9}*864, {6,18}*864, {12,18}*864b
9-fold covers : {18,9}*972a, {18,27}*972, {6,27}*972a, {6,9}*972d, {18,9}*972h, {18,9}*972i, {6,9}*972e, {6,27}*972b, {6,27}*972c, {6,81}*972
10-fold covers : {30,18}*1080a, {6,90}*1080b
11-fold covers : {6,99}*1188
12-fold covers : {18,36}*1296b, {6,36}*1296a, {6,108}*1296b, {36,18}*1296c, {12,18}*1296e, {12,54}*1296b, {6,27}*1296, {12,27}*1296, {18,9}*1296a, {36,9}*1296, {6,9}*1296b, {12,9}*1296c, {6,36}*1296l, {12,18}*1296l
13-fold covers : {6,117}*1404
14-fold covers : {42,18}*1512a, {6,126}*1512b
15-fold covers : {18,45}*1620, {6,45}*1620a, {6,135}*1620
16-fold covers : {6,144}*1728b, {24,36}*1728a, {12,36}*1728b, {24,36}*1728b, {12,72}*1728b, {12,72}*1728d, {48,18}*1728b, {6,9}*1728, {24,9}*1728, {6,36}*1728a, {12,18}*1728a, {6,18}*1728a, {6,36}*1728c, {12,18}*1728b, {12,36}*1728f, {12,36}*1728g, {24,18}*1728b, {24,18}*1728d, {12,18}*1728d, {12,9}*1728, {6,18}*1728c
17-fold covers : {6,153}*1836
18-fold covers : {18,18}*1944b, {18,54}*1944b, {6,54}*1944a, {6,18}*1944h, {18,18}*1944w, {18,18}*1944aa, {6,18}*1944i, {6,54}*1944c, {6,54}*1944e, {6,162}*1944b, {18,18}*1944ad, {18,18}*1944ae, {6,18}*1944m, {6,18}*1944n, {6,18}*1944o, {6,54}*1944g
Permutation Representation (GAP) :
s0 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);;
s1 := ( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,26)
(20,23)(21,24)(25,27);;
s2 := ( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)
(22,25)(23,24)(26,27);;
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*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(27)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);
s1 := Sym(27)!( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)
(18,26)(20,23)(21,24)(25,27);
s2 := Sym(27)!( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)
(17,19)(22,25)(23,24)(26,27);
poly := sub<Sym(27)|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*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope