Polytope of Type {2,9,2}

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

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

to this polytope