Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,47874)
Rank
6
Schläfli Type
{3,2,3,6,6}
Vertices, edges, …
3, 3, 4, 12, 24, 6
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

3-fold

6-fold

Covers minimal covers in bold

None in this atlas.

Representations

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