Polytope of Type {6,18,2,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,2,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {6,18,2,4}
Number of vertices, edges, etc : 6, 54, 18, 4, 4
Order of s₀s₁s₂s₃s₄ : 36
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,18,2,2}*864a
   3-fold quotients : {2,18,2,4}*576, {6,6,2,4}*576a
   6-fold quotients : {2,9,2,4}*288, {2,18,2,2}*288, {6,6,2,2}*288a
   9-fold quotients : {2,6,2,4}*192, {6,2,2,4}*192
   12-fold quotients : {2,9,2,2}*144
   18-fold quotients : {2,3,2,4}*96, {3,2,2,4}*96, {2,6,2,2}*96, {6,2,2,2}*96
   27-fold quotients : {2,2,2,4}*64
   36-fold quotients : {2,3,2,2}*48, {3,2,2,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)
(16,26)(17,25)(18,27)(28,31)(29,33)(30,32)(35,36)(37,50)(38,49)(39,51)(40,47)
(41,46)(42,48)(43,53)(44,52)(45,54);;
s2 := ( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,28)
(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,47)(20,46)(21,48)
(22,50)(23,49)(24,51)(25,53)(26,52)(27,54);;
s3 := (56,57);;
s4 := (55,56)(57,58);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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(58)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);
s1 := Sym(58)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,23)(11,22)(12,24)(13,20)(14,19)
(15,21)(16,26)(17,25)(18,27)(28,31)(29,33)(30,32)(35,36)(37,50)(38,49)(39,51)
(40,47)(41,46)(42,48)(43,53)(44,52)(45,54);
s2 := Sym(58)!( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)
(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,47)(20,46)
(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54);
s3 := Sym(58)!(56,57);
s4 := Sym(58)!(55,56)(57,58);
poly := sub<Sym(58)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s4*s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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