Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*1800b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1800,575)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
150, 450, 150
Order of s0s1s2
30
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

25-fold

50-fold

75-fold

150-fold

225-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)^3> of order 2

78 facets

75 vertex figures

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

75 facets

100 vertex figures

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

50 facets

54 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 5

30 facets

30 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 5

30 facets

30 vertex figures

P/N, where N=<(s0*s1)^3, (s1*s0*s2)^2*s1*s0*(s1*s2)^2> of order 10

18 facets

15 vertex figures

P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, s0*s1*s0*s2*(s1*s0)^2*(s1*s2)^2> of order 10

15 facets

20 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 10

18 facets

15 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle