Part of the Atlas of Small Regular Polytopes

Polytope of Type {30,6}

Atlas Canonical Name {30,6}*1920b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1920,238598)
Rank
3
Schläfli Type
{30,6}
Vertices, edges, …
160, 480, 32
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=<((s1*s0)^2*s1*s2)^2> of order 2

16 facets

80 vertex figures

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

20 facets

80 vertex figures

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

16 facets

80 vertex figures

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

16 facets

80 vertex figures

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

16 facets

80 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

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

12 facets

40 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

Representations

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