Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*648

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(648,547)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
27, 162, 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
  • Non-Orientable

Quotients maximal quotients in bold

3-fold

9-fold

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

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