Polytope of Type {3,2,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,9}*108
if this polytope has a name.
Group : SmallGroup(108,16)
Rank : 4
Schlafli Type : {3,2,9}
Number of vertices, edges, etc : 3, 3, 9, 9
Order of s₀s₁s₂s₃ : 9
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,9,2} of size 216
   {3,2,9,4} of size 432
   {3,2,9,6} of size 648
   {3,2,9,4} of size 864
   {3,2,9,8} of size 1728
   {3,2,9,18} of size 1944
   {3,2,9,6} of size 1944
   {3,2,9,6} of size 1944
   {3,2,9,6} of size 1944
   {3,2,9,6} of size 1944
Vertex Figure Of :
   {2,3,2,9} of size 216
   {3,3,2,9} of size 432
   {4,3,2,9} of size 432
   {6,3,2,9} of size 648
   {4,3,2,9} of size 864
   {6,3,2,9} of size 864
   {5,3,2,9} of size 1080
   {8,3,2,9} of size 1728
   {12,3,2,9} of size 1728
   {6,3,2,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,18}*216, {6,2,9}*216
   3-fold covers : {9,2,9}*324, {3,6,9}*324, {3,2,27}*324
   4-fold covers : {3,2,36}*432, {12,2,9}*432, {6,2,18}*432
   5-fold covers : {3,2,45}*540, {15,2,9}*540
   6-fold covers : {9,2,18}*648, {18,2,9}*648, {3,6,18}*648a, {6,6,9}*648a, {3,2,54}*648, {6,2,27}*648, {3,6,18}*648b, {6,6,9}*648b
   7-fold covers : {3,2,63}*756, {21,2,9}*756
   8-fold covers : {3,2,72}*864, {24,2,9}*864, {6,2,36}*864, {12,2,18}*864, {6,4,18}*864, {3,4,18}*864, {6,4,9}*864
   9-fold covers : {9,6,9}*972, {9,2,27}*972, {27,2,9}*972, {3,6,27}*972, {3,6,9}*972a, {3,6,9}*972b, {3,2,81}*972
   10-fold covers : {3,2,90}*1080, {6,2,45}*1080, {15,2,18}*1080, {30,2,9}*1080
   11-fold covers : {3,2,99}*1188, {33,2,9}*1188
   12-fold covers : {9,2,36}*1296, {36,2,9}*1296, {12,6,9}*1296a, {3,6,36}*1296a, {12,2,27}*1296, {3,2,108}*1296, {18,2,18}*1296, {6,6,18}*1296a, {6,2,54}*1296, {3,6,36}*1296b, {12,6,9}*1296b, {6,6,18}*1296b, {6,6,18}*1296c, {6,6,18}*1296e
   13-fold covers : {3,2,117}*1404, {39,2,9}*1404
   14-fold covers : {3,2,126}*1512, {6,2,63}*1512, {21,2,18}*1512, {42,2,9}*1512
   15-fold covers : {9,2,45}*1620, {45,2,9}*1620, {3,6,45}*1620, {15,6,9}*1620, {3,2,135}*1620, {15,2,27}*1620
   16-fold covers : {3,2,144}*1728, {48,2,9}*1728, {12,2,36}*1728, {12,4,18}*1728, {6,4,36}*1728, {6,2,72}*1728, {24,2,18}*1728, {6,8,18}*1728, {3,4,36}*1728, {3,8,18}*1728, {12,4,9}*1728, {6,8,9}*1728, {3,4,9}*1728, {6,4,18}*1728a, {6,4,18}*1728b
   17-fold covers : {3,2,153}*1836, {51,2,9}*1836
   18-fold covers : {9,6,18}*1944a, {18,6,9}*1944a, {9,2,54}*1944, {18,2,27}*1944, {27,2,18}*1944, {54,2,9}*1944, {3,6,54}*1944a, {6,6,27}*1944a, {3,6,18}*1944a, {6,6,9}*1944a, {3,6,18}*1944b, {6,6,9}*1944b, {3,2,162}*1944, {6,2,81}*1944, {6,18,9}*1944, {9,6,18}*1944b, {18,6,9}*1944b, {3,6,18}*1944c, {3,6,18}*1944d, {6,6,9}*1944c, {6,6,9}*1944d, {3,6,18}*1944e, {6,6,9}*1944e, {3,6,54}*1944b, {6,6,27}*1944b
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7, 8)( 9,10)(11,12);;
s3 := ( 4, 5)( 6, 7)( 8, 9)(10,11);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(12)!(2,3);
s1 := Sym(12)!(1,2);
s2 := Sym(12)!( 5, 6)( 7, 8)( 9,10)(11,12);
s3 := Sym(12)!( 4, 5)( 6, 7)( 8, 9)(10,11);
poly := sub<Sym(12)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope