Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,6,4}

Atlas Canonical Name {10,6,4}*1920a

Overview

Group
SmallGroup(1920,238598)
Rank
4
Schläfli Type
{10,6,4}
Vertices, edges, …
10, 120, 48, 16
Order of s0s1s2s3
30
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

4-fold

5-fold

20-fold

40-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*s2*s3*s2*s1*(s2*s3)^2> of order 2

8 facets

10 vertex figures

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

8 facets

10 vertex figures

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

4 facets

10 vertex figures

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

4 facets

10 vertex figures

Representations

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

References

None.

to this polytope.