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Polytope of Type {2,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,4}*864a
if this polytope has a name.
Group : SmallGroup(864,4007)
Rank : 4
Schlafli Type : {2,6,4}
Number of vertices, edges, etc : 2, 54, 108, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,4,2} of size 1728
Vertex Figure Of :
   {2,2,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,4}*432
   3-fold quotients : {2,6,4}*288
   6-fold quotients : {2,6,4}*144
   27-fold quotients : {2,2,4}*32
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,6,8}*1728a, {4,6,4}*1728a, {2,12,4}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,30)( 4,31)( 5,32)( 6,36)( 7,37)( 8,38)( 9,33)(10,34)(11,35)(12,48)
(13,49)(14,50)(15,54)(16,55)(17,56)(18,51)(19,52)(20,53)(21,39)(22,40)(23,41)
(24,45)(25,46)(26,47)(27,42)(28,43)(29,44);;
s2 := ( 3,12)( 4,14)( 5,13)( 6,16)( 7,15)( 8,17)( 9,20)(10,19)(11,18)(22,23)
(24,25)(27,29)(30,39)(31,41)(32,40)(33,43)(34,42)(35,44)(36,47)(37,46)(38,45)
(49,50)(51,52)(54,56);;
s3 := ( 3, 4)( 6,13)( 7,12)( 8,14)( 9,22)(10,21)(11,23)(15,17)(18,24)(19,26)
(20,25)(27,29)(30,31)(33,40)(34,39)(35,41)(36,49)(37,48)(38,50)(42,44)(45,51)
(46,53)(47,52)(54,56);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(56)!(1,2);
s1 := Sym(56)!( 3,30)( 4,31)( 5,32)( 6,36)( 7,37)( 8,38)( 9,33)(10,34)(11,35)
(12,48)(13,49)(14,50)(15,54)(16,55)(17,56)(18,51)(19,52)(20,53)(21,39)(22,40)
(23,41)(24,45)(25,46)(26,47)(27,42)(28,43)(29,44);
s2 := Sym(56)!( 3,12)( 4,14)( 5,13)( 6,16)( 7,15)( 8,17)( 9,20)(10,19)(11,18)
(22,23)(24,25)(27,29)(30,39)(31,41)(32,40)(33,43)(34,42)(35,44)(36,47)(37,46)
(38,45)(49,50)(51,52)(54,56);
s3 := Sym(56)!( 3, 4)( 6,13)( 7,12)( 8,14)( 9,22)(10,21)(11,23)(15,17)(18,24)
(19,26)(20,25)(27,29)(30,31)(33,40)(34,39)(35,41)(36,49)(37,48)(38,50)(42,44)
(45,51)(46,53)(47,52)(54,56);
poly := sub<Sym(56)|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, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2 >; 
 

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