Part of the Atlas of Small Regular Polytopes

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

Atlas Canonical Name {27,6,2,2}*1296

Overview

Group
SmallGroup(1296,1859)
Rank
5
Schläfli Type
{27,6,2,2}
Vertices, edges, …
27, 81, 6, 2, 2
Order of s0s1s2s3s4
54
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

Covers minimal covers in bold

None in this atlas.

Representations

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