Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,2,6,6,4}

Atlas Canonical Name {3,2,6,6,4}*1728c

Overview

Group
SmallGroup(1728,47409)
Rank
6
Schläfli Type
{3,2,6,6,4}
Vertices, edges, …
3, 3, 6, 18, 12, 4
Order of s0s1s2s3s4s5
12
Order of s0s1s2s3s4s5s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)(25,28)(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)(50,51)(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)(72,74);;
s3 := ( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(28,65)(29,64)(30,66)(31,71)(32,70)(33,72)(34,68)(35,67)(36,69)(37,74)(38,73)(39,75);;
s4 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)(25,37)(26,38)(27,39)(28,34)(29,35)(30,36)(43,46)(44,47)(45,48)(52,55)(53,56)(54,57)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,70)(65,71)(66,72);;
s5 := ( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
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*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s0*s1*s0*s1*s0*s1, 
s3*s4*s5*s4*s3*s4*s5*s4, s4*s5*s4*s5*s4*s5*s4*s5, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s2*s3*s4*s3*s2*s3*s2*s3*s4*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)(25,28)(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)(50,51)(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)(72,74);
s3 := Sym(75)!( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(28,65)(29,64)(30,66)(31,71)(32,70)(33,72)(34,68)(35,67)(36,69)(37,74)(38,73)(39,75);
s4 := Sym(75)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)(25,37)(26,38)(27,39)(28,34)(29,35)(30,36)(43,46)(44,47)(45,48)(52,55)(53,56)(54,57)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,70)(65,71)(66,72);
s5 := Sym(75)!( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75);
poly := sub<Sym(75)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, 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*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s0*s1*s0*s1*s0*s1, s3*s4*s5*s4*s3*s4*s5*s4, 
s4*s5*s4*s5*s4*s5*s4*s5, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s2*s3*s4*s3*s2*s3*s2*s3*s4*s3 >;