Part of the Atlas of Small Regular Polytopes

Polytope of Type {40,6}

Atlas Canonical Name {40,6}*1920c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1920,238599)
Rank
3
Schläfli Type
{40,6}
Vertices, edges, …
160, 480, 24
Order of s0s1s2
30
Order of s0s1s2s1
8
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

16-fold

20-fold

40-fold

48-fold

80-fold

96-fold

160-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*s0*(s2*s1)^2)^2*s0> of order 2

12 facets

80 vertex figures

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

16 facets

80 vertex figures

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

12 facets

80 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

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

12 facets

40 vertex figures

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

8 facets

40 vertex figures

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

8 facets

40 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle