Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*384e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(384,17949)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
32, 96, 32
Order of s0s1s2
8
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

2-fold

4-fold

8-fold

16-fold

48-fold

Covers minimal covers in bold

2-fold

3-fold

5-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*s2)^2*s1)^2> of order 2

16 facets

16 vertex figures

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

16 facets

20 vertex figures

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

16 facets

16 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s2*s1*s2> of order 4

8 facets

8 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2, (s0*(s1*s2)^2*s1)^2> of order 4

8 facets

8 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 4

8 facets

8 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*(s1*s2)^2> of order 4

8 facets

8 vertex figures

P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*(s1*s2)^2*s1)^2> of order 4

8 facets

12 vertex figures

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

8 facets

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5,10)( 6, 9)( 7,11)( 8,12)(15,16)(17,33)(18,34)(19,36)(20,35)(21,42)(22,41)(23,43)(24,44)(25,38)(26,37)(27,39)(28,40)(29,45)(30,46)(31,48)(32,47)(51,52)(53,58)(54,57)(55,59)(56,60)(63,64)(65,81)(66,82)(67,84)(68,83)(69,90)(70,89)(71,91)(72,92)(73,86)(74,85)(75,87)(76,88)(77,93)(78,94)(79,96)(80,95);;
s1 := ( 1,65)( 2,67)( 3,66)( 4,68)( 5,69)( 6,71)( 7,70)( 8,72)( 9,79)(10,77)(11,80)(12,78)(13,74)(14,76)(15,73)(16,75)(17,49)(18,51)(19,50)(20,52)(21,53)(22,55)(23,54)(24,56)(25,63)(26,61)(27,64)(28,62)(29,58)(30,60)(31,57)(32,59)(33,81)(34,83)(35,82)(36,84)(37,85)(38,87)(39,86)(40,88)(41,95)(42,93)(43,96)(44,94)(45,90)(46,92)(47,89)(48,91);;
s2 := ( 1,13)( 2,14)( 3,16)( 4,15)( 5, 6)( 9,10)(17,45)(18,46)(19,48)(20,47)(21,38)(22,37)(23,39)(24,40)(25,42)(26,41)(27,43)(28,44)(29,33)(30,34)(31,36)(32,35)(49,61)(50,62)(51,64)(52,63)(53,54)(57,58)(65,93)(66,94)(67,96)(68,95)(69,86)(70,85)(71,87)(72,88)(73,90)(74,89)(75,91)(76,92)(77,81)(78,82)(79,84)(80,83);;
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(96)!( 3, 4)( 5,10)( 6, 9)( 7,11)( 8,12)(15,16)(17,33)(18,34)(19,36)(20,35)(21,42)(22,41)(23,43)(24,44)(25,38)(26,37)(27,39)(28,40)(29,45)(30,46)(31,48)(32,47)(51,52)(53,58)(54,57)(55,59)(56,60)(63,64)(65,81)(66,82)(67,84)(68,83)(69,90)(70,89)(71,91)(72,92)(73,86)(74,85)(75,87)(76,88)(77,93)(78,94)(79,96)(80,95);
s1 := Sym(96)!( 1,65)( 2,67)( 3,66)( 4,68)( 5,69)( 6,71)( 7,70)( 8,72)( 9,79)(10,77)(11,80)(12,78)(13,74)(14,76)(15,73)(16,75)(17,49)(18,51)(19,50)(20,52)(21,53)(22,55)(23,54)(24,56)(25,63)(26,61)(27,64)(28,62)(29,58)(30,60)(31,57)(32,59)(33,81)(34,83)(35,82)(36,84)(37,85)(38,87)(39,86)(40,88)(41,95)(42,93)(43,96)(44,94)(45,90)(46,92)(47,89)(48,91);
s2 := Sym(96)!( 1,13)( 2,14)( 3,16)( 4,15)( 5, 6)( 9,10)(17,45)(18,46)(19,48)(20,47)(21,38)(22,37)(23,39)(24,40)(25,42)(26,41)(27,43)(28,44)(29,33)(30,34)(31,36)(32,35)(49,61)(50,62)(51,64)(52,63)(53,54)(57,58)(65,93)(66,94)(67,96)(68,95)(69,86)(70,85)(71,87)(72,88)(73,90)(74,89)(75,91)(76,92)(77,81)(78,82)(79,84)(80,83);
poly := sub<Sym(96)|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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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