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Polytope of Type {14,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,2}*336
if this polytope has a name.
Group : SmallGroup(336,219)
Rank : 4
Schlafli Type : {14,6,2}
Number of vertices, edges, etc : 14, 42, 6, 2
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,6,2,2} of size 672
   {14,6,2,3} of size 1008
   {14,6,2,4} of size 1344
   {14,6,2,5} of size 1680
Vertex Figure Of :
   {2,14,6,2} of size 672
   {4,14,6,2} of size 1344
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {14,2,2}*112
   6-fold quotients : {7,2,2}*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 : {14,12,2}*672, {28,6,2}*672a, {14,6,4}*672a
   3-fold covers : {14,18,2}*1008, {14,6,6}*1008a, {14,6,6}*1008c, {42,6,2}*1008a, {42,6,2}*1008b
   4-fold covers : {28,6,4}*1344a, {14,12,4}*1344a, {14,24,2}*1344, {56,6,2}*1344, {14,6,8}*1344, {28,12,2}*1344, {14,6,4}*1344, {28,6,2}*1344
   5-fold covers : {14,6,10}*1680, {14,30,2}*1680, {70,6,2}*1680
Permutation Representation (GAP) :
s0 := ( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)
(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)(46,47)
(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(72,77)(73,76)
(74,75)(79,84)(80,83)(81,82);;
s1 := ( 1,44)( 2,43)( 3,49)( 4,48)( 5,47)( 6,46)( 7,45)( 8,58)( 9,57)(10,63)
(11,62)(12,61)(13,60)(14,59)(15,51)(16,50)(17,56)(18,55)(19,54)(20,53)(21,52)
(22,65)(23,64)(24,70)(25,69)(26,68)(27,67)(28,66)(29,79)(30,78)(31,84)(32,83)
(33,82)(34,81)(35,80)(36,72)(37,71)(38,77)(39,76)(40,75)(41,74)(42,73);;
s2 := ( 1,71)( 2,72)( 3,73)( 4,74)( 5,75)( 6,76)( 7,77)( 8,64)( 9,65)(10,66)
(11,67)(12,68)(13,69)(14,70)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)
(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,43)(30,44)(31,45)(32,46)
(33,47)(34,48)(35,49)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63);;
s3 := (85,86);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(86)!( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)
(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)
(46,47)(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(72,77)
(73,76)(74,75)(79,84)(80,83)(81,82);
s1 := Sym(86)!( 1,44)( 2,43)( 3,49)( 4,48)( 5,47)( 6,46)( 7,45)( 8,58)( 9,57)
(10,63)(11,62)(12,61)(13,60)(14,59)(15,51)(16,50)(17,56)(18,55)(19,54)(20,53)
(21,52)(22,65)(23,64)(24,70)(25,69)(26,68)(27,67)(28,66)(29,79)(30,78)(31,84)
(32,83)(33,82)(34,81)(35,80)(36,72)(37,71)(38,77)(39,76)(40,75)(41,74)(42,73);
s2 := Sym(86)!( 1,71)( 2,72)( 3,73)( 4,74)( 5,75)( 6,76)( 7,77)( 8,64)( 9,65)
(10,66)(11,67)(12,68)(13,69)(14,70)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)
(21,84)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,43)(30,44)(31,45)
(32,46)(33,47)(34,48)(35,49)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63);
s3 := Sym(86)!(85,86);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 

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