Polytope of Type {2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18}*432a
if this polytope has a name.
Group : SmallGroup(432,544)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 6, 54, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,18,2} of size 864
   {2,6,18,4} of size 1728
   {2,6,18,4} of size 1728
Vertex Figure Of :
   {2,2,6,18} of size 864
   {3,2,6,18} of size 1296
   {4,2,6,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,18}*144, {2,6,6}*144a
   6-fold quotients : {2,2,9}*72
   9-fold quotients : {2,2,6}*48, {2,6,2}*48
   18-fold quotients : {2,2,3}*24, {2,3,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,6,36}*864a, {2,12,18}*864a, {4,6,18}*864a
   3-fold covers : {2,18,18}*1296a, {2,6,18}*1296b, {2,6,54}*1296a, {6,6,18}*1296b, {6,6,18}*1296d, {2,6,18}*1296i
   4-fold covers : {4,6,36}*1728a, {4,12,18}*1728a, {2,6,72}*1728a, {2,24,18}*1728a, {8,6,18}*1728a, {2,12,36}*1728a, {4,6,18}*1728, {2,6,36}*1728, {2,12,18}*1728a
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 6, 9)( 7,10)( 8,11)(15,18)(16,19)(17,20)(24,27)(25,28)(26,29)(33,36)
(34,37)(35,38)(42,45)(43,46)(44,47)(51,54)(52,55)(53,56);;
s2 := ( 3, 6)( 4, 8)( 5, 7)(10,11)(12,25)(13,24)(14,26)(15,22)(16,21)(17,23)
(18,28)(19,27)(20,29)(30,33)(31,35)(32,34)(37,38)(39,52)(40,51)(41,53)(42,49)
(43,48)(44,50)(45,55)(46,54)(47,56);;
s3 := ( 3,39)( 4,41)( 5,40)( 6,42)( 7,44)( 8,43)( 9,45)(10,47)(11,46)(12,30)
(13,32)(14,31)(15,33)(16,35)(17,34)(18,36)(19,38)(20,37)(21,49)(22,48)(23,50)
(24,52)(25,51)(26,53)(27,55)(28,54)(29,56);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*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(56)!(1,2);
s1 := Sym(56)!( 6, 9)( 7,10)( 8,11)(15,18)(16,19)(17,20)(24,27)(25,28)(26,29)
(33,36)(34,37)(35,38)(42,45)(43,46)(44,47)(51,54)(52,55)(53,56);
s2 := Sym(56)!( 3, 6)( 4, 8)( 5, 7)(10,11)(12,25)(13,24)(14,26)(15,22)(16,21)
(17,23)(18,28)(19,27)(20,29)(30,33)(31,35)(32,34)(37,38)(39,52)(40,51)(41,53)
(42,49)(43,48)(44,50)(45,55)(46,54)(47,56);
s3 := Sym(56)!( 3,39)( 4,41)( 5,40)( 6,42)( 7,44)( 8,43)( 9,45)(10,47)(11,46)
(12,30)(13,32)(14,31)(15,33)(16,35)(17,34)(18,36)(19,38)(20,37)(21,49)(22,48)
(23,50)(24,52)(25,51)(26,53)(27,55)(28,54)(29,56);
poly := sub<Sym(56)|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, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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