Polytope of Type {14,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6}*168
Also Known As : {14,6|2}. if this polytope has another name.
Group : SmallGroup(168,50)
Rank : 3
Schlafli Type : {14,6}
Number of vertices, edges, etc : 14, 42, 6
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,6,2} of size 336
   {14,6,3} of size 504
   {14,6,4} of size 672
   {14,6,3} of size 672
   {14,6,4} of size 672
   {14,6,6} of size 1008
   {14,6,6} of size 1008
   {14,6,6} of size 1008
   {14,6,8} of size 1344
   {14,6,4} of size 1344
   {14,6,6} of size 1344
   {14,6,9} of size 1512
   {14,6,3} of size 1512
   {14,6,5} of size 1680
   {14,6,5} of size 1680
   {14,6,10} of size 1680
Vertex Figure Of :
   {2,14,6} of size 336
   {4,14,6} of size 672
   {6,14,6} of size 1008
   {7,14,6} of size 1176
   {8,14,6} of size 1344
   {10,14,6} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {14,2}*56
   6-fold quotients : {7,2}*28
   7-fold quotients : {2,6}*24
   14-fold quotients : {2,3}*12
   21-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,12}*336, {28,6}*336a
   3-fold covers : {14,18}*504, {42,6}*504a, {42,6}*504b
   4-fold covers : {14,24}*672, {56,6}*672, {28,12}*672, {28,6}*672
   5-fold covers : {14,30}*840, {70,6}*840
   6-fold covers : {14,36}*1008, {28,18}*1008a, {84,6}*1008a, {42,12}*1008a, {42,12}*1008b, {84,6}*1008b
   7-fold covers : {98,6}*1176, {14,42}*1176a, {14,42}*1176b
   8-fold covers : {14,48}*1344, {112,6}*1344, {28,12}*1344a, {28,24}*1344a, {56,12}*1344a, {28,24}*1344b, {56,12}*1344b, {28,12}*1344b, {28,6}*1344e, {56,6}*1344b, {56,6}*1344c, {28,12}*1344c
   9-fold covers : {14,54}*1512, {42,18}*1512a, {42,6}*1512a, {126,6}*1512a, {42,18}*1512b, {42,6}*1512c, {42,6}*1512d
   10-fold covers : {14,60}*1680, {28,30}*1680a, {70,12}*1680, {140,6}*1680a
   11-fold covers : {14,66}*1848, {154,6}*1848
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);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, 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(84)!( 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(84)!( 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(84)!( 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);
poly := sub<Sym(84)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

References : None.
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