Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,47874)
Rank
6
Schläfli Type
{3,2,4,6,6}
Vertices, edges, …
3, 3, 4, 12, 18, 6
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 := ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75);;
s3 := ( 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);;
s4 := ( 4,16)( 5,17)( 6,19)( 7,18)( 8,24)( 9,25)(10,27)(11,26)(12,20)(13,21)(14,23)(15,22)(30,31)(32,36)(33,37)(34,39)(35,38)(40,52)(41,53)(42,55)(43,54)(44,60)(45,61)(46,63)(47,62)(48,56)(49,57)(50,59)(51,58)(66,67)(68,72)(69,73)(70,75)(71,74);;
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*s2*s3, s3*s4*s5*s4*s3*s4*s5*s4, 
s2*s3*s4*s3*s2*s3*s4*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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)!( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75);
s3 := 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);
s4 := Sym(75)!( 4,16)( 5,17)( 6,19)( 7,18)( 8,24)( 9,25)(10,27)(11,26)(12,20)(13,21)(14,23)(15,22)(30,31)(32,36)(33,37)(34,39)(35,38)(40,52)(41,53)(42,55)(43,54)(44,60)(45,61)(46,63)(47,62)(48,56)(49,57)(50,59)(51,58)(66,67)(68,72)(69,73)(70,75)(71,74);
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*s2*s3, 
s3*s4*s5*s4*s3*s4*s5*s4, s2*s3*s4*s3*s2*s3*s4*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;