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

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

s0 := (1,2);;
s1 := ( 4, 9)( 5, 8)( 6, 7)(11,16)(12,15)(13,14)(18,23)(19,22)(20,21)(25,30)
(26,29)(27,28)(32,37)(33,36)(34,35)(39,44)(40,43)(41,42)(46,51)(47,50)(48,49)
(53,58)(54,57)(55,56)(60,65)(61,64)(62,63)(67,72)(68,71)(69,70)(74,79)(75,78)
(76,77)(81,86)(82,85)(83,84);;
s2 := ( 3, 4)( 5, 9)( 6, 8)(10,11)(12,16)(13,15)(17,18)(19,23)(20,22)(24,25)
(26,30)(27,29)(31,32)(33,37)(34,36)(38,39)(40,44)(41,43)(45,67)(46,66)(47,72)
(48,71)(49,70)(50,69)(51,68)(52,74)(53,73)(54,79)(55,78)(56,77)(57,76)(58,75)
(59,81)(60,80)(61,86)(62,85)(63,84)(64,83)(65,82);;
s3 := ( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,50)( 9,51)(10,59)(11,60)(12,61)
(13,62)(14,63)(15,64)(16,65)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)
(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,80)(32,81)(33,82)(34,83)
(35,84)(36,85)(37,86)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79);;
s4 := ( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)( 8,15)( 9,16)(24,31)(25,32)(26,33)
(27,34)(28,35)(29,36)(30,37)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)(51,58)
(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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(86)!(1,2);
s1 := Sym(86)!( 4, 9)( 5, 8)( 6, 7)(11,16)(12,15)(13,14)(18,23)(19,22)(20,21)
(25,30)(26,29)(27,28)(32,37)(33,36)(34,35)(39,44)(40,43)(41,42)(46,51)(47,50)
(48,49)(53,58)(54,57)(55,56)(60,65)(61,64)(62,63)(67,72)(68,71)(69,70)(74,79)
(75,78)(76,77)(81,86)(82,85)(83,84);
s2 := Sym(86)!( 3, 4)( 5, 9)( 6, 8)(10,11)(12,16)(13,15)(17,18)(19,23)(20,22)
(24,25)(26,30)(27,29)(31,32)(33,37)(34,36)(38,39)(40,44)(41,43)(45,67)(46,66)
(47,72)(48,71)(49,70)(50,69)(51,68)(52,74)(53,73)(54,79)(55,78)(56,77)(57,76)
(58,75)(59,81)(60,80)(61,86)(62,85)(63,84)(64,83)(65,82);
s3 := Sym(86)!( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,50)( 9,51)(10,59)(11,60)
(12,61)(13,62)(14,63)(15,64)(16,65)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)
(23,58)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,80)(32,81)(33,82)
(34,83)(35,84)(36,85)(37,86)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79);
s4 := Sym(86)!( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)( 8,15)( 9,16)(24,31)(25,32)
(26,33)(27,34)(28,35)(29,36)(30,37)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)
(51,58)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79);
poly := sub<Sym(86)|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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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