Polytope of Type {6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,14}*168
Also Known As : {6,14|2}. if this polytope has another name.
Group : SmallGroup(168,50)
Rank : 3
Schlafli Type : {6,14}
Number of vertices, edges, etc : 6, 42, 14
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 :
   {6,14,2} of size 336
   {6,14,4} of size 672
   {6,14,6} of size 1008
   {6,14,7} of size 1176
   {6,14,8} of size 1344
   {6,14,10} of size 1680
Vertex Figure Of :
   {2,6,14} of size 336
   {3,6,14} of size 504
   {4,6,14} of size 672
   {3,6,14} of size 672
   {4,6,14} of size 672
   {6,6,14} of size 1008
   {6,6,14} of size 1008
   {6,6,14} of size 1008
   {8,6,14} of size 1344
   {4,6,14} of size 1344
   {6,6,14} of size 1344
   {9,6,14} of size 1512
   {3,6,14} of size 1512
   {5,6,14} of size 1680
   {5,6,14} of size 1680
   {10,6,14} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,14}*56
   6-fold quotients : {2,7}*28
   7-fold quotients : {6,2}*24
   14-fold quotients : {3,2}*12
   21-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,14}*336, {6,28}*336a
   3-fold covers : {18,14}*504, {6,42}*504a, {6,42}*504b
   4-fold covers : {24,14}*672, {6,56}*672, {12,28}*672, {6,28}*672
   5-fold covers : {30,14}*840, {6,70}*840
   6-fold covers : {36,14}*1008, {18,28}*1008a, {6,84}*1008a, {12,42}*1008a, {12,42}*1008b, {6,84}*1008b
   7-fold covers : {6,98}*1176, {42,14}*1176a, {42,14}*1176b
   8-fold covers : {48,14}*1344, {6,112}*1344, {12,28}*1344a, {24,28}*1344a, {12,56}*1344a, {24,28}*1344b, {12,56}*1344b, {12,28}*1344b, {6,28}*1344e, {6,56}*1344b, {6,56}*1344c, {12,28}*1344c
   9-fold covers : {54,14}*1512, {18,42}*1512a, {6,42}*1512a, {6,126}*1512a, {18,42}*1512b, {6,42}*1512c, {6,42}*1512d
   10-fold covers : {60,14}*1680, {30,28}*1680a, {12,70}*1680, {6,140}*1680a
   11-fold covers : {66,14}*1848, {6,154}*1848
Permutation Representation (GAP) :

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

References : None.
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