Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,47409)
Rank
6
Schläfli Type
{3,2,4,6,6}
Vertices, edges, …
3, 3, 4, 12, 18, 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

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