Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3}

Atlas Canonical Name {6,3}*324

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(324,41)
Rank
3
Schläfli Type
{6,3}
Vertices, edges, …
54, 81, 27
Order of s0s1s2
18
Order of s0s1s2s1
6
Also known as
{6,3}(3,3). if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2> of order 3

9 facets

18 vertex figures

P/N, where N=<(s0*s1)^2> of order 3

11 facets

18 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)(16,24)(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)(42,52)(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)(68,81)(69,79)(70,78)(71,76)(72,77);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27)(28,66)(29,65)(30,64)(31,69)(32,68)(33,67)(34,72)(35,71)(36,70)(37,57)(38,56)(39,55)(40,60)(41,59)(42,58)(43,63)(44,62)(45,61)(46,75)(47,74)(48,73)(49,78)(50,77)(51,76)(52,81)(53,80)(54,79);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,33)( 5,32)( 6,31)( 7,35)( 8,34)( 9,36)(10,53)(11,52)(12,54)(13,46)(14,48)(15,47)(16,51)(17,50)(18,49)(19,40)(20,42)(21,41)(22,45)(23,44)(24,43)(25,38)(26,37)(27,39)(55,57)(58,59)(62,63)(64,79)(65,81)(66,80)(67,75)(68,74)(69,73)(70,77)(71,76)(72,78);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(81)!( 4, 8)( 5, 9)( 6, 7)(10,19)(11,20)(12,21)(13,26)(14,27)(15,25)(16,24)(17,22)(18,23)(31,35)(32,36)(33,34)(37,46)(38,47)(39,48)(40,53)(41,54)(42,52)(43,51)(44,49)(45,50)(58,62)(59,63)(60,61)(64,73)(65,74)(66,75)(67,80)(68,81)(69,79)(70,78)(71,76)(72,77);
s1 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27)(28,66)(29,65)(30,64)(31,69)(32,68)(33,67)(34,72)(35,71)(36,70)(37,57)(38,56)(39,55)(40,60)(41,59)(42,58)(43,63)(44,62)(45,61)(46,75)(47,74)(48,73)(49,78)(50,77)(51,76)(52,81)(53,80)(54,79);
s2 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,33)( 5,32)( 6,31)( 7,35)( 8,34)( 9,36)(10,53)(11,52)(12,54)(13,46)(14,48)(15,47)(16,51)(17,50)(18,49)(19,40)(20,42)(21,41)(22,45)(23,44)(24,43)(25,38)(26,37)(27,39)(55,57)(58,59)(62,63)(64,79)(65,81)(66,80)(67,75)(68,74)(69,73)(70,77)(71,76)(72,78);
poly := sub<Sym(81)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle