Part of the Atlas of Small Regular Polytopes

Polytope of Type {30,6}

Atlas Canonical Name {30,6}*1920c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1920,238599)
Rank
3
Schläfli Type
{30,6}
Vertices, edges, …
160, 480, 32
Order of s0s1s2
40
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=<(s0*s1)^2*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 2

20 facets

80 vertex figures

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

16 facets

80 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

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

12 facets

40 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

Representations

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

References

None.

to this polytope.

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