Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*648

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Overview

Group
SmallGroup(648,547)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
54, 162, 27
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
  • Self-Petrie

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

References

None.

to this polytope.

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