Polytope of Type {2,2,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,12}*1728f
if this polytope has a name.
Group : SmallGroup(1728,46139)
Rank : 5
Schlafli Type : {2,2,6,12}
Number of vertices, edges, etc : 2, 2, 18, 108, 36
Order of s₀s₁s₂s₃s₄ : 4
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 : {2,2,6,12}*864b
   3-fold quotients : {2,2,6,4}*576
   6-fold quotients : {2,2,6,4}*288
   27-fold quotients : {2,2,2,4}*64
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

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