Polytope of Type {2,39,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,39,4}*624
if this polytope has a name.
Group : SmallGroup(624,245)
Rank : 4
Schlafli Type : {2,39,4}
Number of vertices, edges, etc : 2, 39, 78, 4
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,39,4,2} of size 1248
Vertex Figure Of :
   {2,2,39,4} of size 1248
   {3,2,39,4} of size 1872
Quotients (Maximal Quotients in Boldface) :
   13-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,39,4}*1248, {2,78,4}*1248b, {2,78,4}*1248c
   3-fold covers : {2,117,4}*1872, {6,39,4}*1872
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 5, 6)( 7,51)( 8,52)( 9,54)(10,53)(11,47)(12,48)(13,50)(14,49)(15,43)
(16,44)(17,46)(18,45)(19,39)(20,40)(21,42)(22,41)(23,35)(24,36)(25,38)(26,37)
(27,31)(28,32)(29,34)(30,33);;
s2 := ( 3, 7)( 4, 9)( 5, 8)( 6,10)(11,51)(12,53)(13,52)(14,54)(15,47)(16,49)
(17,48)(18,50)(19,43)(20,45)(21,44)(22,46)(23,39)(24,41)(25,40)(26,42)(27,35)
(28,37)(29,36)(30,38)(32,33);;
s3 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)
(45,46)(47,48)(49,50)(51,52)(53,54);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*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*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)!( 5, 6)( 7,51)( 8,52)( 9,54)(10,53)(11,47)(12,48)(13,50)(14,49)
(15,43)(16,44)(17,46)(18,45)(19,39)(20,40)(21,42)(22,41)(23,35)(24,36)(25,38)
(26,37)(27,31)(28,32)(29,34)(30,33);
s2 := Sym(54)!( 3, 7)( 4, 9)( 5, 8)( 6,10)(11,51)(12,53)(13,52)(14,54)(15,47)
(16,49)(17,48)(18,50)(19,43)(20,45)(21,44)(22,46)(23,39)(24,41)(25,40)(26,42)
(27,35)(28,37)(29,36)(30,38)(32,33);
s3 := Sym(54)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46)(47,48)(49,50)(51,52)(53,54);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*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*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 >;

to this polytope