Polytope of Type {2,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,14}*336
if this polytope has a name.
Group : SmallGroup(336,219)
Rank : 4
Schlafli Type : {2,6,14}
Number of vertices, edges, etc : 2, 6, 42, 14
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,14,2} of size 672
   {2,6,14,4} of size 1344
Vertex Figure Of :
   {2,2,6,14} of size 672
   {3,2,6,14} of size 1008
   {4,2,6,14} of size 1344
   {5,2,6,14} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,14}*112
   6-fold quotients : {2,2,7}*56
   7-fold quotients : {2,6,2}*48
   14-fold quotients : {2,3,2}*24
   21-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,12,14}*672, {2,6,28}*672a, {4,6,14}*672a
   3-fold covers : {2,18,14}*1008, {6,6,14}*1008a, {6,6,14}*1008b, {2,6,42}*1008a, {2,6,42}*1008b
   4-fold covers : {4,6,28}*1344a, {4,12,14}*1344a, {2,24,14}*1344, {2,6,56}*1344, {8,6,14}*1344, {2,12,28}*1344, {4,6,14}*1344, {2,6,28}*1344
   5-fold covers : {10,6,14}*1680, {2,30,14}*1680, {2,6,70}*1680
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*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*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

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

to this polytope