Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296j

▶ 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=<s0*s1*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s1*s2> of order 2

27 facets

54 vertex figures

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

30 facets

54 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

References

None.

to this polytope.

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