Polytope of Type {4,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,28}*224
Also Known As : {4,28|2}. if this polytope has another name.
Group : SmallGroup(224,77)
Rank : 3
Schlafli Type : {4,28}
Number of vertices, edges, etc : 4, 56, 28
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,28,2} of size 448
   {4,28,4} of size 896
   {4,28,6} of size 1344
   {4,28,8} of size 1792
   {4,28,8} of size 1792
   {4,28,4} of size 1792
Vertex Figure Of :
   {2,4,28} of size 448
   {4,4,28} of size 896
   {6,4,28} of size 1344
   {3,4,28} of size 1344
   {8,4,28} of size 1792
   {8,4,28} of size 1792
   {4,4,28} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,28}*112, {4,14}*112
   4-fold quotients : {2,14}*56
   7-fold quotients : {4,4}*32
   8-fold quotients : {2,7}*28
   14-fold quotients : {2,4}*16, {4,2}*16
   28-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,56}*448a, {4,28}*448, {4,56}*448b, {8,28}*448a, {8,28}*448b
   3-fold covers : {12,28}*672, {4,84}*672a
   4-fold covers : {4,56}*896a, {8,56}*896a, {8,56}*896b, {8,28}*896a, {8,56}*896c, {8,56}*896d, {4,112}*896a, {4,112}*896b, {4,28}*896, {4,56}*896b, {8,28}*896b, {16,28}*896a, {16,28}*896b
   5-fold covers : {20,28}*1120, {4,140}*1120
   6-fold covers : {12,28}*1344a, {24,28}*1344a, {12,56}*1344a, {24,28}*1344b, {12,56}*1344b, {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b
   7-fold covers : {4,196}*1568, {28,28}*1568a, {28,28}*1568b
   8-fold covers : {8,56}*1792a, {8,28}*1792a, {8,56}*1792b, {4,56}*1792a, {8,56}*1792c, {8,56}*1792d, {16,28}*1792a, {4,112}*1792a, {16,28}*1792b, {4,112}*1792b, {8,112}*1792a, {16,56}*1792a, {8,112}*1792b, {16,56}*1792b, {16,56}*1792c, {8,112}*1792c, {8,112}*1792d, {16,56}*1792d, {16,56}*1792e, {8,112}*1792e, {8,112}*1792f, {16,56}*1792f, {32,28}*1792a, {4,224}*1792a, {32,28}*1792b, {4,224}*1792b, {4,28}*1792, {4,56}*1792b, {8,28}*1792b, {8,28}*1792c, {8,56}*1792e, {4,56}*1792c, {4,56}*1792d, {8,28}*1792d, {8,56}*1792f, {8,56}*1792g, {8,56}*1792h
Permutation Representation (GAP) :

s0 := (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)
(46,53)(47,54)(48,55)(49,56);;
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)(15,43)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)
(22,50)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51);;
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)(29,44)(30,43)(31,49)(32,48)(33,47)(34,46)(35,45)(36,51)(37,50)
(38,56)(39,55)(40,54)(41,53)(42,52);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*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*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)!(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)
(45,52)(46,53)(47,54)(48,55)(49,56);
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)(15,43)(16,49)(17,48)(18,47)(19,46)(20,45)
(21,44)(22,50)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51);
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)(29,44)(30,43)(31,49)(32,48)(33,47)(34,46)(35,45)(36,51)
(37,50)(38,56)(39,55)(40,54)(41,53)(42,52);
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*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*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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