Part of the Atlas of Small Regular Polytopes

Polytope of Type {40,4}

Atlas Canonical Name {40,4}*320b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(320,374)
Rank
3
Schläfli Type
{40,4}
Vertices, edges, …
40, 80, 4
Order of s0s1s2
40
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

4-fold

5-fold

8-fold

10-fold

16-fold

20-fold

40-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

Twisty Puzzle