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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,12,2}*1728i
if this polytope has a name.
Group : SmallGroup(1728,47912)
Rank : 5
Schlafli Type : {2,6,12,2}
Number of vertices, edges, etc : 2, 18, 108, 36, 2
Order of s₀s₁s₂s₃s₄ : 12
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 : {2,6,12,2}*864c
   3-fold quotients : {2,6,4,2}*576
   6-fold quotients : {2,6,4,2}*288
   9-fold quotients : {2,2,12,2}*192
   18-fold quotients : {2,2,6,2}*96
   27-fold quotients : {2,2,4,2}*64
   36-fold quotients : {2,2,3,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

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