Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,4,20}

Atlas Canonical Name {3,4,20}*1920

Overview

Group
SmallGroup(1920,238598)
Rank
4
Schläfli Type
{3,4,20}
Vertices, edges, …
6, 24, 160, 40
Order of s0s1s2s3
30
Order of s0s1s2s3s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Orientable
  • Flat

Quotients maximal quotients in bold

4-fold

5-fold

10-fold

16-fold

20-fold

32-fold

40-fold

80-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*s3*s2)^2*s1> of order 2

20 facets

6 vertex figures

P/N, where N=<s0*(s1*s0*s2)^2*s1, s0*s1*s0*s3*s2*s1*s0*s2*s1*s3> of order 4

20 facets

  • 20 of 2-fold non-regular quotient of {3,4}*48

4 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)( 5,13)( 6,16)( 7,15)( 8,14)(10,12)(18,20)(21,29)(22,32)(23,31)(24,30)(26,28)(34,36)(37,45)(38,48)(39,47)(40,46)(42,44)(50,52)(53,61)(54,64)(55,63)(56,62)(58,60)(66,68)(69,77)(70,80)(71,79)(72,78)(74,76);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,77)(26,78)(27,79)(28,80)(29,73)(30,74)(31,75)(32,76)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60);;
s3 := ( 1,17)( 2,18)( 3,19)( 4,20)( 5,22)( 6,21)( 7,24)( 8,23)( 9,27)(10,28)(11,25)(12,26)(13,32)(14,31)(15,30)(16,29)(33,65)(34,66)(35,67)(36,68)(37,70)(38,69)(39,72)(40,71)(41,75)(42,76)(43,73)(44,74)(45,80)(46,79)(47,78)(48,77)(53,54)(55,56)(57,59)(58,60)(61,64)(62,63);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1, 
s1*s2*s3*s2*s1*s3*s0*s2*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s0 ];;
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)( 5,13)( 6,16)( 7,15)( 8,14)(10,12)(18,20)(21,29)(22,32)(23,31)(24,30)(26,28)(34,36)(37,45)(38,48)(39,47)(40,46)(42,44)(50,52)(53,61)(54,64)(55,63)(56,62)(58,60)(66,68)(69,77)(70,80)(71,79)(72,78)(74,76);
s2 := Sym(80)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,77)(26,78)(27,79)(28,80)(29,73)(30,74)(31,75)(32,76)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60);
s3 := Sym(80)!( 1,17)( 2,18)( 3,19)( 4,20)( 5,22)( 6,21)( 7,24)( 8,23)( 9,27)(10,28)(11,25)(12,26)(13,32)(14,31)(15,30)(16,29)(33,65)(34,66)(35,67)(36,68)(37,70)(38,69)(39,72)(40,71)(41,75)(42,76)(43,73)(44,74)(45,80)(46,79)(47,78)(48,77)(53,54)(55,56)(57,59)(58,60)(61,64)(62,63);
poly := sub<Sym(80)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1, 
s1*s2*s3*s2*s1*s3*s0*s2*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s0 >;

References

None.

to this polytope.