Polytope of Type {2,18,6}

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

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