Polytope of Type {2,54}

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

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

to this polytope