Polytope of Type {6,2,4,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,4,14}*1344
if this polytope has a name.
Group : SmallGroup(1344,11527)
Rank : 5
Schlafli Type : {6,2,4,14}
Number of vertices, edges, etc : 6, 6, 4, 28, 14
Order of s₀s₁s₂s₃s₄ : 84
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 : {3,2,4,14}*672, {6,2,2,14}*672
   3-fold quotients : {2,2,4,14}*448
   4-fold quotients : {3,2,2,14}*336, {6,2,2,7}*336
   6-fold quotients : {2,2,2,14}*224
   7-fold quotients : {6,2,4,2}*192
   8-fold quotients : {3,2,2,7}*168
   12-fold quotients : {2,2,2,7}*112
   14-fold quotients : {3,2,4,2}*96, {6,2,2,2}*96
   21-fold quotients : {2,2,4,2}*64
   28-fold quotients : {3,2,2,2}*48
   42-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 8,11)(12,17)(13,18)(19,25)(20,26)(27,31)(28,32);;
s3 := ( 7, 8)( 9,13)(10,12)(11,16)(14,20)(15,19)(17,24)(18,23)(21,28)(22,27)
(25,30)(26,29)(31,34)(32,33);;
s4 := ( 7, 9)( 8,12)(10,14)(11,17)(13,19)(15,21)(16,23)(18,25)(20,27)(24,29)
(26,31)(30,33);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(34)!(3,4)(5,6);
s1 := Sym(34)!(1,5)(2,3)(4,6);
s2 := Sym(34)!( 8,11)(12,17)(13,18)(19,25)(20,26)(27,31)(28,32);
s3 := Sym(34)!( 7, 8)( 9,13)(10,12)(11,16)(14,20)(15,19)(17,24)(18,23)(21,28)
(22,27)(25,30)(26,29)(31,34)(32,33);
s4 := Sym(34)!( 7, 9)( 8,12)(10,14)(11,17)(13,19)(15,21)(16,23)(18,25)(20,27)
(24,29)(26,31)(30,33);
poly := sub<Sym(34)|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, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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