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Polytope of Type {8,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,14}*224
Also Known As : {8,14|2}. if this polytope has another name.
Group : SmallGroup(224,105)
Rank : 3
Schlafli Type : {8,14}
Number of vertices, edges, etc : 8, 56, 14
Order of s0s1s2 : 56
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,14,2} of size 448
   {8,14,4} of size 896
   {8,14,6} of size 1344
   {8,14,7} of size 1568
   {8,14,8} of size 1792
Vertex Figure Of :
   {2,8,14} of size 448
   {4,8,14} of size 896
   {4,8,14} of size 896
   {6,8,14} of size 1344
   {3,8,14} of size 1344
   {4,8,14} of size 1792
   {8,8,14} of size 1792
   {8,8,14} of size 1792
   {8,8,14} of size 1792
   {8,8,14} of size 1792
   {4,8,14} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,14}*112
   4-fold quotients : {2,14}*56
   7-fold quotients : {8,2}*32
   8-fold quotients : {2,7}*28
   14-fold quotients : {4,2}*16
   28-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,28}*448a, {16,14}*448
   3-fold covers : {24,14}*672, {8,42}*672
   4-fold covers : {8,56}*896b, {8,28}*896a, {8,56}*896d, {16,28}*896a, {16,28}*896b, {32,14}*896
   5-fold covers : {40,14}*1120, {8,70}*1120
   6-fold covers : {48,14}*1344, {24,28}*1344a, {8,84}*1344a, {16,42}*1344
   7-fold covers : {8,98}*1568, {56,14}*1568a, {56,14}*1568c
   8-fold covers : {8,56}*1792a, {8,28}*1792a, {8,56}*1792c, {16,28}*1792a, {16,28}*1792b, {8,112}*1792a, {8,112}*1792b, {16,56}*1792c, {8,112}*1792d, {16,56}*1792d, {16,56}*1792e, {8,112}*1792f, {16,56}*1792f, {32,28}*1792a, {32,28}*1792b, {64,14}*1792
Permutation Representation (GAP) :
s0 := (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43)(30,44)(31,45)
(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)
(42,56);;
s1 := ( 1,29)( 2,35)( 3,34)( 4,33)( 5,32)( 6,31)( 7,30)( 8,36)( 9,42)(10,41)
(11,40)(12,39)(13,38)(14,37)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)
(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44);;
s2 := ( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,16)(17,21)(18,20)(22,23)
(24,28)(25,27)(29,30)(31,35)(32,34)(36,37)(38,42)(39,41)(43,44)(45,49)(46,48)
(50,51)(52,56)(53,55);;
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*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(56)!(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43)(30,44)
(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)
(42,56);
s1 := Sym(56)!( 1,29)( 2,35)( 3,34)( 4,33)( 5,32)( 6,31)( 7,30)( 8,36)( 9,42)
(10,41)(11,40)(12,39)(13,38)(14,37)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)
(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44);
s2 := Sym(56)!( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,16)(17,21)(18,20)
(22,23)(24,28)(25,27)(29,30)(31,35)(32,34)(36,37)(38,42)(39,41)(43,44)(45,49)
(46,48)(50,51)(52,56)(53,55);
poly := sub<Sym(56)|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*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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