Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*288a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(288,1028)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
24, 72, 24
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

2-fold

3-fold

4-fold

6-fold

8-fold

12-fold

24-fold

36-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-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)^2*s0*s2*s1*s0*s1*s2> of order 2

12 facets

12 vertex figures

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

16 facets

12 vertex figures

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

12 facets

12 vertex figures

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

12 facets

8 vertex figures

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

6 facets

6 vertex figures

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

6 facets

6 vertex figures

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

6 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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