Polytope of Type {9,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4}*72
if this polytope has a name.
Group : SmallGroup(72,15)
Rank : 3
Schlafli Type : {9,4}
Number of vertices, edges, etc : 9, 18, 4
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {9,4,2} of size 144
   {9,4,4} of size 576
   {9,4,4} of size 1152
Vertex Figure Of :
   {2,9,4} of size 144
   {4,9,4} of size 288
   {6,9,4} of size 432
   {4,9,4} of size 576
   {8,9,4} of size 1152
   {18,9,4} of size 1296
   {6,9,4} of size 1296
   {6,9,4} of size 1296
   {6,9,4} of size 1296
   {6,9,4} of size 1296
   {6,9,4} of size 1728
   {12,9,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,4}*144, {18,4}*144b, {18,4}*144c
   3-fold covers : {27,4}*216
   4-fold covers : {36,4}*288b, {36,4}*288c, {9,8}*288, {18,4}*288
   5-fold covers : {45,4}*360
   6-fold covers : {27,4}*432, {54,4}*432b, {54,4}*432c, {9,12}*432, {18,12}*432c
   7-fold covers : {63,4}*504
   8-fold covers : {18,4}*576a, {9,8}*576, {18,8}*576a, {72,4}*576c, {72,4}*576d, {36,4}*576b, {18,4}*576b, {36,4}*576c, {18,8}*576b, {18,8}*576c
   9-fold covers : {81,4}*648
   10-fold covers : {18,20}*720b, {45,4}*720, {90,4}*720b, {90,4}*720c
   11-fold covers : {99,4}*792
   12-fold covers : {108,4}*864b, {108,4}*864c, {27,8}*864, {54,4}*864, {9,24}*864, {18,12}*864a, {18,12}*864b
   13-fold covers : {117,4}*936
   14-fold covers : {18,28}*1008b, {63,4}*1008, {126,4}*1008b, {126,4}*1008c
   15-fold covers : {135,4}*1080
   16-fold covers : {36,4}*1152b, {36,4}*1152c, {9,8}*1152, {18,8}*1152a, {36,8}*1152c, {36,8}*1152d, {18,8}*1152b, {18,8}*1152c, {144,4}*1152c, {144,4}*1152d, {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {18,4}*1152a, {18,8}*1152d, {18,8}*1152e, {18,8}*1152f, {36,8}*1152g, {36,8}*1152h, {72,4}*1152c, {72,4}*1152d, {18,8}*1152g, {36,4}*1152e, {72,4}*1152e, {18,4}*1152b, {72,4}*1152f
   17-fold covers : {153,4}*1224
   18-fold covers : {81,4}*1296, {162,4}*1296b, {162,4}*1296c, {27,12}*1296, {54,12}*1296c, {9,36}*1296, {18,36}*1296d, {9,12}*1296c, {18,12}*1296k
   19-fold covers : {171,4}*1368
   20-fold covers : {180,4}*1440b, {180,4}*1440c, {45,8}*1440, {18,20}*1440, {90,4}*1440
   21-fold covers : {189,4}*1512
   22-fold covers : {18,44}*1584b, {99,4}*1584, {198,4}*1584b, {198,4}*1584c
   23-fold covers : {207,4}*1656
   24-fold covers : {54,4}*1728a, {27,8}*1728, {54,8}*1728a, {216,4}*1728c, {216,4}*1728d, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c, {9,24}*1728, {18,24}*1728a, {36,12}*1728e, {36,12}*1728f, {18,12}*1728c, {36,12}*1728g, {18,24}*1728b, {18,24}*1728c, {18,24}*1728d, {18,24}*1728e, {18,12}*1728d, {36,12}*1728h, {9,12}*1728, {36,12}*1728i
   25-fold covers : {225,4}*1800
   26-fold covers : {18,52}*1872b, {117,4}*1872, {234,4}*1872b, {234,4}*1872c
   27-fold covers : {243,4}*1944, {9,4}*1944, {9,12}*1944a, {9,12}*1944b, {9,12}*1944c
Permutation Representation (GAP) :

s0 := ( 1, 2)( 3, 6)( 4, 5)( 7,15)( 8,14)( 9,16)(10,12)(11,13)(17,23)(18,24)
(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);;
s1 := ( 1, 5)( 2, 3)( 4,12)( 6, 8)( 7, 9)(10,21)(11,22)(13,15)(14,17)(16,18)
(19,29)(20,30)(23,25)(24,26)(27,31)(28,35)(32,33)(34,36);;
s2 := ( 1,15)( 2, 7)( 3, 9)( 6,16)(10,20)(12,22)(17,26)(19,28)(21,30)(23,32)
(25,33)(31,36);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*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(36)!( 1, 2)( 3, 6)( 4, 5)( 7,15)( 8,14)( 9,16)(10,12)(11,13)(17,23)
(18,24)(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);
s1 := Sym(36)!( 1, 5)( 2, 3)( 4,12)( 6, 8)( 7, 9)(10,21)(11,22)(13,15)(14,17)
(16,18)(19,29)(20,30)(23,25)(24,26)(27,31)(28,35)(32,33)(34,36);
s2 := Sym(36)!( 1,15)( 2, 7)( 3, 9)( 6,16)(10,20)(12,22)(17,26)(19,28)(21,30)
(23,32)(25,33)(31,36);
poly := sub<Sym(36)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*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