Polytope of Type {9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*108
if this polytope has a name.
Group : SmallGroup(108,16)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 9, 27, 6
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 :
   {9,6,2} of size 216
   {9,6,3} of size 324
   {9,6,4} of size 432
   {9,6,6} of size 648
   {9,6,6} of size 648
   {9,6,8} of size 864
   {9,6,9} of size 972
   {9,6,3} of size 972
   {9,6,10} of size 1080
   {9,6,12} of size 1296
   {9,6,12} of size 1296
   {9,6,4} of size 1296
   {9,6,14} of size 1512
   {9,6,15} of size 1620
   {9,6,16} of size 1728
   {9,6,4} of size 1728
   {9,6,18} of size 1944
   {9,6,6} of size 1944
   {9,6,18} of size 1944
   {9,6,6} of size 1944
   {9,6,6} of size 1944
Vertex Figure Of :
   {2,9,6} of size 216
   {4,9,6} of size 432
   {6,9,6} of size 648
   {4,9,6} of size 864
   {8,9,6} of size 1728
   {18,9,6} 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
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,2}*36, {3,6}*36
   9-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,6}*216b
   3-fold covers : {9,18}*324, {9,6}*324a, {27,6}*324
   4-fold covers : {36,6}*432b, {18,12}*432b, {9,6}*432, {9,12}*432
   5-fold covers : {45,6}*540
   6-fold covers : {18,18}*648c, {18,6}*648a, {54,6}*648b, {18,6}*648i
   7-fold covers : {63,6}*756
   8-fold covers : {72,6}*864b, {36,12}*864b, {18,24}*864b, {9,12}*864, {9,24}*864, {18,6}*864, {18,12}*864b
   9-fold covers : {9,18}*972a, {27,18}*972, {27,6}*972a, {9,6}*972d, {9,18}*972h, {9,18}*972i, {9,6}*972e, {27,6}*972b, {27,6}*972c, {81,6}*972
   10-fold covers : {18,30}*1080a, {90,6}*1080b
   11-fold covers : {99,6}*1188
   12-fold covers : {36,18}*1296b, {36,6}*1296a, {108,6}*1296b, {18,36}*1296c, {18,12}*1296e, {54,12}*1296b, {27,6}*1296, {27,12}*1296, {9,18}*1296a, {9,36}*1296, {9,6}*1296b, {9,12}*1296c, {36,6}*1296l, {18,12}*1296l
   13-fold covers : {117,6}*1404
   14-fold covers : {18,42}*1512a, {126,6}*1512b
   15-fold covers : {45,18}*1620, {45,6}*1620a, {135,6}*1620
   16-fold covers : {144,6}*1728b, {36,24}*1728a, {36,12}*1728b, {36,24}*1728b, {72,12}*1728b, {72,12}*1728d, {18,48}*1728b, {9,6}*1728, {9,24}*1728, {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, {9,12}*1728, {18,6}*1728b
   17-fold covers : {153,6}*1836
   18-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, 5)( 6, 9)( 7,11)( 8,10)(12,15)(13,17)(14,16)(18,21)(19,23)
(20,22)(24,27)(25,26);;
s1 := ( 1, 7)( 2, 4)( 3,13)( 5, 8)( 6,10)( 9,19)(11,14)(12,16)(15,24)(17,20)
(18,22)(21,26)(23,25);;
s2 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(27)!( 2, 3)( 4, 5)( 6, 9)( 7,11)( 8,10)(12,15)(13,17)(14,16)(18,21)
(19,23)(20,22)(24,27)(25,26);
s1 := Sym(27)!( 1, 7)( 2, 4)( 3,13)( 5, 8)( 6,10)( 9,19)(11,14)(12,16)(15,24)
(17,20)(18,22)(21,26)(23,25);
s2 := Sym(27)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(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*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 >;

References : None.
to this polytope