Polytope of Type {3,2,6,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,27}*1944
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 5
Schlafli Type : {3,2,6,27}
Number of vertices, edges, etc : 3, 3, 6, 81, 27
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 : {3,2,2,27}*648, {3,2,6,9}*648
   9-fold quotients : {3,2,2,9}*216, {3,2,6,3}*216
   27-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope