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

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

s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := ( 5,47)( 6,48)( 7,49)( 8,50)( 9,51)(10,52)(11,53)(12,61)(13,62)(14,63)
(15,64)(16,65)(17,66)(18,67)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)
(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,82)(34,83)(35,84)(36,85)
(37,86)(38,87)(39,88)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81);;
s3 := ( 5,54)( 6,60)( 7,59)( 8,58)( 9,57)(10,56)(11,55)(12,47)(13,53)(14,52)
(15,51)(16,50)(17,49)(18,48)(19,61)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)
(26,75)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,68)(34,74)(35,73)(36,72)
(37,71)(38,70)(39,69)(40,82)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83);;
s4 := ( 5,27)( 6,26)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,34)(13,33)(14,39)
(15,38)(16,37)(17,36)(18,35)(19,41)(20,40)(21,46)(22,45)(23,44)(24,43)(25,42)
(47,69)(48,68)(49,74)(50,73)(51,72)(52,71)(53,70)(54,76)(55,75)(56,81)(57,80)
(58,79)(59,78)(60,77)(61,83)(62,82)(63,88)(64,87)(65,86)(66,85)(67,84);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(88)!(2,3);
s1 := Sym(88)!(1,2)(3,4);
s2 := Sym(88)!( 5,47)( 6,48)( 7,49)( 8,50)( 9,51)(10,52)(11,53)(12,61)(13,62)
(14,63)(15,64)(16,65)(17,66)(18,67)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)
(25,60)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,82)(34,83)(35,84)
(36,85)(37,86)(38,87)(39,88)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81);
s3 := Sym(88)!( 5,54)( 6,60)( 7,59)( 8,58)( 9,57)(10,56)(11,55)(12,47)(13,53)
(14,52)(15,51)(16,50)(17,49)(18,48)(19,61)(20,67)(21,66)(22,65)(23,64)(24,63)
(25,62)(26,75)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,68)(34,74)(35,73)
(36,72)(37,71)(38,70)(39,69)(40,82)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83);
s4 := Sym(88)!( 5,27)( 6,26)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,34)(13,33)
(14,39)(15,38)(16,37)(17,36)(18,35)(19,41)(20,40)(21,46)(22,45)(23,44)(24,43)
(25,42)(47,69)(48,68)(49,74)(50,73)(51,72)(52,71)(53,70)(54,76)(55,75)(56,81)
(57,80)(58,79)(59,78)(60,77)(61,83)(62,82)(63,88)(64,87)(65,86)(66,85)(67,84);
poly := sub<Sym(88)|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, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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 >;

to this polytope