Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,47874)
Rank
6
Schläfli Type
{3,2,6,6,4}
Vertices, edges, …
3, 3, 6, 18, 12, 4
Order of s0s1s2s3s4s5
6
Order of s0s1s2s3s4s5s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

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