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Polytope of Type {2,26,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,26,4}*416
if this polytope has a name.
Group : SmallGroup(416,216)
Rank : 4
Schlafli Type : {2,26,4}
Number of vertices, edges, etc : 2, 26, 52, 4
Order of s0s1s2s3 : 52
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,26,4,2} of size 832
   {2,26,4,4} of size 1664
Vertex Figure Of :
   {2,2,26,4} of size 832
   {3,2,26,4} of size 1248
   {4,2,26,4} of size 1664
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,26,2}*208
   4-fold quotients : {2,13,2}*104
   13-fold quotients : {2,2,4}*32
   26-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,52,4}*832, {4,26,4}*832, {2,26,8}*832
   3-fold covers : {2,26,12}*1248, {6,26,4}*1248, {2,78,4}*1248a
   4-fold covers : {4,52,4}*1664, {2,52,8}*1664a, {2,104,4}*1664a, {2,52,8}*1664b, {2,104,4}*1664b, {2,52,4}*1664, {4,26,8}*1664, {8,26,4}*1664, {2,26,16}*1664
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(17,28)(18,27)(19,26)(20,25)
(21,24)(22,23)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,54)(44,53)(45,52)
(46,51)(47,50)(48,49);;
s2 := ( 3, 4)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)(16,17)(18,28)(19,27)(20,26)
(21,25)(22,24)(29,43)(30,42)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)
(38,47)(39,46)(40,45)(41,44);;
s3 := ( 3,29)( 4,30)( 5,31)( 6,32)( 7,33)( 8,34)( 9,35)(10,36)(11,37)(12,38)
(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)
(24,50)(25,51)(26,52)(27,53)(28,54);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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(54)!(1,2);
s1 := Sym(54)!( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(17,28)(18,27)(19,26)
(20,25)(21,24)(22,23)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,54)(44,53)
(45,52)(46,51)(47,50)(48,49);
s2 := Sym(54)!( 3, 4)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)(16,17)(18,28)(19,27)
(20,26)(21,25)(22,24)(29,43)(30,42)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)
(37,48)(38,47)(39,46)(40,45)(41,44);
s3 := Sym(54)!( 3,29)( 4,30)( 5,31)( 6,32)( 7,33)( 8,34)( 9,35)(10,36)(11,37)
(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)
(23,49)(24,50)(25,51)(26,52)(27,53)(28,54);
poly := sub<Sym(54)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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 >; 
 

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