Polytope of Type {28,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,4}*224
Also Known As : {28,4|2}. if this polytope has another name.
Group : SmallGroup(224,77)
Rank : 3
Schlafli Type : {28,4}
Number of vertices, edges, etc : 28, 56, 4
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {28,4,2} of size 448
   {28,4,4} of size 896
   {28,4,6} of size 1344
   {28,4,3} of size 1344
   {28,4,8} of size 1792
   {28,4,8} of size 1792
   {28,4,4} of size 1792
Vertex Figure Of :
   {2,28,4} of size 448
   {4,28,4} of size 896
   {6,28,4} of size 1344
   {8,28,4} of size 1792
   {8,28,4} of size 1792
   {4,28,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,2}*112, {14,4}*112
   4-fold quotients : {14,2}*56
   7-fold quotients : {4,4}*32
   8-fold quotients : {7,2}*28
   14-fold quotients : {2,4}*16, {4,2}*16
   28-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {56,4}*448a, {28,4}*448, {56,4}*448b, {28,8}*448a, {28,8}*448b
   3-fold covers : {28,12}*672, {84,4}*672a
   4-fold covers : {56,4}*896a, {56,8}*896a, {56,8}*896b, {28,8}*896a, {56,8}*896c, {56,8}*896d, {112,4}*896a, {112,4}*896b, {28,4}*896, {56,4}*896b, {28,8}*896b, {28,16}*896a, {28,16}*896b
   5-fold covers : {28,20}*1120, {140,4}*1120
   6-fold covers : {28,12}*1344a, {28,24}*1344a, {56,12}*1344a, {28,24}*1344b, {56,12}*1344b, {168,4}*1344a, {84,4}*1344a, {168,4}*1344b, {84,8}*1344a, {84,8}*1344b
   7-fold covers : {196,4}*1568, {28,28}*1568a, {28,28}*1568c
   8-fold covers : {56,8}*1792a, {28,8}*1792a, {56,8}*1792b, {56,4}*1792a, {56,8}*1792c, {56,8}*1792d, {28,16}*1792a, {112,4}*1792a, {28,16}*1792b, {112,4}*1792b, {112,8}*1792a, {56,16}*1792a, {112,8}*1792b, {56,16}*1792b, {56,16}*1792c, {112,8}*1792c, {112,8}*1792d, {56,16}*1792d, {56,16}*1792e, {112,8}*1792e, {112,8}*1792f, {56,16}*1792f, {28,32}*1792a, {224,4}*1792a, {28,32}*1792b, {224,4}*1792b, {28,4}*1792, {56,4}*1792b, {28,8}*1792b, {28,8}*1792c, {56,8}*1792e, {56,4}*1792c, {56,4}*1792d, {28,8}*1792d, {56,8}*1792f, {56,8}*1792g, {56,8}*1792h
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)(29,43)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,50)(37,56)
(38,55)(39,54)(40,53)(41,52)(42,51);;
s1 := ( 1,30)( 2,29)( 3,35)( 4,34)( 5,33)( 6,32)( 7,31)( 8,37)( 9,36)(10,42)
(11,41)(12,40)(13,39)(14,38)(15,44)(16,43)(17,49)(18,48)(19,47)(20,46)(21,45)
(22,51)(23,50)(24,56)(25,55)(26,54)(27,53)(28,52);;
s2 := (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);;
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, 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*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(56)!( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)
(23,28)(24,27)(25,26)(29,43)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,50)
(37,56)(38,55)(39,54)(40,53)(41,52)(42,51);
s1 := Sym(56)!( 1,30)( 2,29)( 3,35)( 4,34)( 5,33)( 6,32)( 7,31)( 8,37)( 9,36)
(10,42)(11,41)(12,40)(13,39)(14,38)(15,44)(16,43)(17,49)(18,48)(19,47)(20,46)
(21,45)(22,51)(23,50)(24,56)(25,55)(26,54)(27,53)(28,52);
s2 := 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);
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, 
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*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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