Polytope of Type {2,2,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,9}*72
if this polytope has a name.
Group : SmallGroup(72,17)
Rank : 4
Schlafli Type : {2,2,9}
Number of vertices, edges, etc : 2, 2, 9, 9
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,2,9,2} of size 144
   {2,2,9,4} of size 288
   {2,2,9,6} of size 432
   {2,2,9,4} of size 576
   {2,2,9,8} of size 1152
   {2,2,9,18} of size 1296
   {2,2,9,6} of size 1296
   {2,2,9,6} of size 1296
   {2,2,9,6} of size 1296
   {2,2,9,6} of size 1296
   {2,2,9,6} of size 1728
   {2,2,9,12} of size 1728
Vertex Figure Of :
   {2,2,2,9} of size 144
   {3,2,2,9} of size 216
   {4,2,2,9} of size 288
   {5,2,2,9} of size 360
   {6,2,2,9} of size 432
   {7,2,2,9} of size 504
   {8,2,2,9} of size 576
   {9,2,2,9} of size 648
   {10,2,2,9} of size 720
   {11,2,2,9} of size 792
   {12,2,2,9} of size 864
   {13,2,2,9} of size 936
   {14,2,2,9} of size 1008
   {15,2,2,9} of size 1080
   {16,2,2,9} of size 1152
   {17,2,2,9} of size 1224
   {18,2,2,9} of size 1296
   {19,2,2,9} of size 1368
   {20,2,2,9} of size 1440
   {21,2,2,9} of size 1512
   {22,2,2,9} of size 1584
   {23,2,2,9} of size 1656
   {24,2,2,9} of size 1728
   {25,2,2,9} of size 1800
   {26,2,2,9} of size 1872
   {27,2,2,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,2,9}*144, {2,2,18}*144
   3-fold covers : {2,2,27}*216, {2,6,9}*216, {6,2,9}*216
   4-fold covers : {8,2,9}*288, {2,2,36}*288, {2,4,18}*288a, {4,2,18}*288, {2,4,9}*288
   5-fold covers : {10,2,9}*360, {2,2,45}*360
   6-fold covers : {4,2,27}*432, {2,2,54}*432, {12,2,9}*432, {4,6,9}*432, {2,6,18}*432a, {2,6,18}*432b, {6,2,18}*432
   7-fold covers : {14,2,9}*504, {2,2,63}*504
   8-fold covers : {16,2,9}*576, {2,4,36}*576a, {4,2,36}*576, {4,4,18}*576, {2,2,72}*576, {2,8,18}*576, {8,2,18}*576, {4,4,9}*576b, {2,8,9}*576, {2,4,18}*576
   9-fold covers : {2,2,81}*648, {2,18,9}*648, {18,2,9}*648, {6,6,9}*648a, {2,6,9}*648a, {2,6,27}*648, {6,2,27}*648, {6,6,9}*648b
   10-fold covers : {20,2,9}*720, {4,2,45}*720, {2,10,18}*720, {10,2,18}*720, {2,2,90}*720
   11-fold covers : {22,2,9}*792, {2,2,99}*792
   12-fold covers : {8,2,27}*864, {2,2,108}*864, {2,4,54}*864a, {4,2,54}*864, {24,2,9}*864, {8,6,9}*864, {2,4,27}*864, {2,6,36}*864a, {2,6,36}*864b, {6,2,36}*864, {2,12,18}*864a, {12,2,18}*864, {4,6,18}*864a, {6,4,18}*864, {4,6,18}*864b, {2,12,18}*864b, {2,6,9}*864, {6,4,9}*864, {2,12,9}*864
   13-fold covers : {26,2,9}*936, {2,2,117}*936
   14-fold covers : {28,2,9}*1008, {4,2,63}*1008, {2,14,18}*1008, {14,2,18}*1008, {2,2,126}*1008
   15-fold covers : {10,2,27}*1080, {2,2,135}*1080, {10,6,9}*1080, {2,6,45}*1080, {6,2,45}*1080, {30,2,9}*1080
   16-fold covers : {32,2,9}*1152, {4,4,36}*1152, {4,8,18}*1152a, {8,4,18}*1152a, {2,8,36}*1152a, {2,4,72}*1152a, {4,8,18}*1152b, {8,4,18}*1152b, {2,8,36}*1152b, {2,4,72}*1152b, {4,4,18}*1152a, {2,4,36}*1152a, {8,2,36}*1152, {4,2,72}*1152, {2,16,18}*1152, {16,2,18}*1152, {2,2,144}*1152, {4,4,9}*1152b, {2,8,9}*1152, {4,8,9}*1152, {8,4,9}*1152, {2,4,36}*1152b, {4,4,18}*1152d, {2,4,18}*1152b, {2,4,36}*1152c, {2,8,18}*1152b, {2,8,18}*1152c
   17-fold covers : {34,2,9}*1224, {2,2,153}*1224
   18-fold covers : {4,2,81}*1296, {2,2,162}*1296, {36,2,9}*1296, {12,6,9}*1296a, {12,2,27}*1296, {4,18,9}*1296, {4,6,9}*1296a, {4,6,27}*1296, {2,18,18}*1296a, {2,18,18}*1296b, {18,2,18}*1296, {6,6,18}*1296a, {2,6,18}*1296a, {2,6,18}*1296b, {2,6,54}*1296a, {2,6,54}*1296b, {6,2,54}*1296, {12,6,9}*1296b, {4,6,9}*1296e, {6,6,18}*1296b, {6,6,18}*1296c, {6,6,18}*1296d, {6,6,18}*1296e, {2,6,18}*1296i
   19-fold covers : {38,2,9}*1368, {2,2,171}*1368
   20-fold covers : {40,2,9}*1440, {8,2,45}*1440, {2,10,36}*1440, {10,2,36}*1440, {2,20,18}*1440a, {20,2,18}*1440, {4,10,18}*1440, {10,4,18}*1440, {2,2,180}*1440, {2,4,90}*1440a, {4,2,90}*1440, {10,4,9}*1440, {2,4,45}*1440
   21-fold covers : {14,2,27}*1512, {2,2,189}*1512, {14,6,9}*1512, {2,6,63}*1512, {6,2,63}*1512, {42,2,9}*1512
   22-fold covers : {44,2,9}*1584, {4,2,99}*1584, {2,22,18}*1584, {22,2,18}*1584, {2,2,198}*1584
   23-fold covers : {46,2,9}*1656, {2,2,207}*1656
   24-fold covers : {16,2,27}*1728, {2,4,108}*1728a, {4,2,108}*1728, {4,4,54}*1728, {2,2,216}*1728, {2,8,54}*1728, {8,2,54}*1728, {48,2,9}*1728, {16,6,9}*1728, {4,4,27}*1728b, {2,8,27}*1728, {12,2,36}*1728, {4,6,36}*1728a, {4,12,18}*1728a, {12,4,18}*1728, {6,4,36}*1728, {2,6,72}*1728a, {2,6,72}*1728b, {6,2,72}*1728, {2,24,18}*1728a, {24,2,18}*1728, {6,8,18}*1728, {8,6,18}*1728a, {2,12,36}*1728a, {2,12,36}*1728b, {4,6,36}*1728b, {8,6,18}*1728b, {2,24,18}*1728b, {4,12,18}*1728b, {2,4,54}*1728, {2,12,9}*1728, {4,6,9}*1728a, {12,4,9}*1728, {2,24,9}*1728, {6,8,9}*1728, {4,12,9}*1728, {4,6,18}*1728, {6,4,18}*1728a, {6,6,18}*1728, {2,6,18}*1728, {2,6,36}*1728, {6,4,18}*1728b, {2,12,18}*1728a, {2,12,18}*1728b
   25-fold covers : {50,2,9}*1800, {2,2,225}*1800, {2,10,9}*1800, {2,10,45}*1800, {10,2,45}*1800
   26-fold covers : {52,2,9}*1872, {4,2,117}*1872, {2,26,18}*1872, {26,2,18}*1872, {2,2,234}*1872
   27-fold covers : {2,2,243}*1944, {18,6,9}*1944a, {2,18,9}*1944a, {2,18,27}*1944, {18,2,27}*1944, {54,2,9}*1944, {6,6,27}*1944a, {2,6,27}*1944a, {6,6,9}*1944a, {2,6,9}*1944d, {2,18,9}*1944h, {2,18,9}*1944i, {6,6,9}*1944b, {2,6,9}*1944e, {2,6,27}*1944b, {2,6,27}*1944c, {2,6,81}*1944, {6,2,81}*1944, {6,18,9}*1944, {18,6,9}*1944b, {6,6,9}*1944c, {6,6,9}*1944d, {6,6,9}*1944e, {6,6,27}*1944b
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8, 9)(10,11)(12,13);;
s3 := ( 5, 6)( 7, 8)( 9,10)(11,12);;
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*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, 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(13)!(1,2);
s1 := Sym(13)!(3,4);
s2 := Sym(13)!( 6, 7)( 8, 9)(10,11)(12,13);
s3 := Sym(13)!( 5, 6)( 7, 8)( 9,10)(11,12);
poly := sub<Sym(13)|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*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope