Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,30}

Atlas Canonical Name {6,30}*1920b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1920,238598)
Rank
3
Schläfli Type
{6,30}
Vertices, edges, …
32, 480, 160
Order of s0s1s2
20
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

4-fold

5-fold

8-fold

10-fold

20-fold

40-fold

48-fold

80-fold

96-fold

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

80 facets

16 vertex figures

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

80 facets

20 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

40 facets

8 vertex figures

P/N, where N=<(s0*(s1*s2)^2*s1)^2, s0*s1*s0*(s2*s1)^3*s0*(s2*s1)^2> of order 4

40 facets

8 vertex figures

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

40 facets

12 vertex figures

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

40 facets

8 vertex figures

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

40 facets

8 vertex figures

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

40 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle