Polytope of Type {54,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,2}*216
if this polytope has a name.
Group : SmallGroup(216,23)
Rank : 3
Schlafli Type : {54,2}
Number of vertices, edges, etc : 54, 54, 2
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {54,2,2} of size 432
   {54,2,3} of size 648
   {54,2,4} of size 864
   {54,2,5} of size 1080
   {54,2,6} of size 1296
   {54,2,7} of size 1512
   {54,2,8} of size 1728
   {54,2,9} of size 1944
Vertex Figure Of :
   {2,54,2} of size 432
   {4,54,2} of size 864
   {4,54,2} of size 864
   {4,54,2} of size 864
   {6,54,2} of size 1296
   {6,54,2} of size 1296
   {8,54,2} of size 1728
   {4,54,2} of size 1728
   {6,54,2} of size 1944
   {6,54,2} of size 1944
   {6,54,2} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {27,2}*108
   3-fold quotients : {18,2}*72
   6-fold quotients : {9,2}*36
   9-fold quotients : {6,2}*24
   18-fold quotients : {3,2}*12
   27-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {108,2}*432, {54,4}*432a
   3-fold covers : {162,2}*648, {54,6}*648a, {54,6}*648b
   4-fold covers : {108,4}*864a, {216,2}*864, {54,8}*864, {54,4}*864
   5-fold covers : {54,10}*1080, {270,2}*1080
   6-fold covers : {324,2}*1296, {162,4}*1296a, {54,12}*1296a, {108,6}*1296a, {108,6}*1296b, {54,12}*1296b
   7-fold covers : {54,14}*1512, {378,2}*1512
   8-fold covers : {216,4}*1728a, {108,4}*1728a, {216,4}*1728b, {108,8}*1728a, {108,8}*1728b, {432,2}*1728, {54,16}*1728, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c
   9-fold covers : {486,2}*1944, {54,18}*1944a, {54,18}*1944b, {54,6}*1944a, {54,6}*1944b, {162,6}*1944a, {162,6}*1944b, {54,6}*1944g
Permutation Representation (GAP) :

s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)
(45,46)(47,48)(49,50)(51,52)(53,54);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,21)(18,19)
(20,25)(22,23)(24,29)(26,27)(28,33)(30,31)(32,37)(34,35)(36,41)(38,39)(40,45)
(42,43)(44,49)(46,47)(48,53)(50,51)(52,54);;
s2 := (55,56);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(56)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46)(47,48)(49,50)(51,52)(53,54);
s1 := Sym(56)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,21)
(18,19)(20,25)(22,23)(24,29)(26,27)(28,33)(30,31)(32,37)(34,35)(36,41)(38,39)
(40,45)(42,43)(44,49)(46,47)(48,53)(50,51)(52,54);
s2 := Sym(56)!(55,56);
poly := sub<Sym(56)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope