Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296k

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,2909)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

108-fold

162-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

27 facets

54 vertex figures

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

30 facets

54 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

P/N, where N=<(s0*s2*s1)^4> of order 3

18 facets

36 vertex figures

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

18 facets

36 vertex figures

P/N, where N=<(s0*(s1*s2)^2*s1)^2, (s0*s1)^3*s0*s2*(s1*s0)^2*s2*s1> of order 6

12 facets

18 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^6> of order 6

12 facets

18 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 9

6 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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