Polytope of Type {2,2,27,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,27,6}*1296
if this polytope has a name.
Group : SmallGroup(1296,1859)
Rank : 5
Schlafli Type : {2,2,27,6}
Number of vertices, edges, etc : 2, 2, 27, 81, 6
Order of s₀s₁s₂s₃s₄ : 54
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,27,2}*432, {2,2,9,6}*432
   9-fold quotients : {2,2,9,2}*144, {2,2,3,6}*144
   27-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8,12)( 9,11)(10,13)(14,23)(15,25)(16,24)(17,30)(18,29)(19,31)
(20,27)(21,26)(22,28)(32,62)(33,64)(34,63)(35,59)(36,61)(37,60)(38,66)(39,65)
(40,67)(41,80)(42,82)(43,81)(44,77)(45,79)(46,78)(47,84)(48,83)(49,85)(50,71)
(51,73)(52,72)(53,68)(54,70)(55,69)(56,75)(57,74)(58,76);;
s3 := ( 5,41)( 6,43)( 7,42)( 8,48)( 9,47)(10,49)(11,45)(12,44)(13,46)(14,32)
(15,34)(16,33)(17,39)(18,38)(19,40)(20,36)(21,35)(22,37)(23,50)(24,52)(25,51)
(26,57)(27,56)(28,58)(29,54)(30,53)(31,55)(59,71)(60,73)(61,72)(62,68)(63,70)
(64,69)(65,75)(66,74)(67,76)(77,80)(78,82)(79,81)(83,84);;
s4 := (14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(41,50)
(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(68,77)(69,78)(70,79)
(71,80)(72,81)(73,82)(74,83)(75,84)(76,85);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 6, 7)( 8,12)( 9,11)(10,13)(14,23)(15,25)(16,24)(17,30)(18,29)
(19,31)(20,27)(21,26)(22,28)(32,62)(33,64)(34,63)(35,59)(36,61)(37,60)(38,66)
(39,65)(40,67)(41,80)(42,82)(43,81)(44,77)(45,79)(46,78)(47,84)(48,83)(49,85)
(50,71)(51,73)(52,72)(53,68)(54,70)(55,69)(56,75)(57,74)(58,76);
s3 := Sym(85)!( 5,41)( 6,43)( 7,42)( 8,48)( 9,47)(10,49)(11,45)(12,44)(13,46)
(14,32)(15,34)(16,33)(17,39)(18,38)(19,40)(20,36)(21,35)(22,37)(23,50)(24,52)
(25,51)(26,57)(27,56)(28,58)(29,54)(30,53)(31,55)(59,71)(60,73)(61,72)(62,68)
(63,70)(64,69)(65,75)(66,74)(67,76)(77,80)(78,82)(79,81)(83,84);
s4 := Sym(85)!(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)
(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(68,77)(69,78)
(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(76,85);
poly := sub<Sym(85)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope