Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,10}

Atlas Canonical Name {4,6,10}*1440

Overview

Group
SmallGroup(1440,5890)
Rank
4
Schläfli Type
{4,6,10}
Vertices, edges, …
12, 36, 90, 10
Order of s0s1s2s3
20
Order of s0s1s2s3s2s1
2
Also known as
{{4,6}4,{6,10|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

5-fold

9-fold

10-fold

18-fold

36-fold

45-fold

90-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*s2*s1*s0*s2> of order 3

10 facets

  • 10 of 3-fold non-regular quotient of {4,6}*144

8 vertex figures

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

10 facets

  • 10 of 3-fold non-regular quotient of {4,6}*144

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 6,36)( 7,37)( 8,38)( 9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(51,81)(52,82)(53,83)(54,84)(55,85)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80);;
s1 := (16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,86)(62,87)(63,88)(64,89)(65,90)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)(73,83)(74,84)(75,85);;
s2 := ( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,16)( 7,20)( 8,19)( 9,18)(10,17)(11,26)(12,30)(13,29)(14,28)(15,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(46,66)(47,70)(48,69)(49,68)(50,67)(51,61)(52,65)(53,64)(54,63)(55,62)(56,71)(57,75)(58,74)(59,73)(60,72)(76,81)(77,85)(78,84)(79,83)(80,82)(87,90)(88,89);;
s3 := ( 1,47)( 2,46)( 3,50)( 4,49)( 5,48)( 6,52)( 7,51)( 8,55)( 9,54)(10,53)(11,57)(12,56)(13,60)(14,59)(15,58)(16,62)(17,61)(18,65)(19,64)(20,63)(21,67)(22,66)(23,70)(24,69)(25,68)(26,72)(27,71)(28,75)(29,74)(30,73)(31,77)(32,76)(33,80)(34,79)(35,78)(36,82)(37,81)(38,85)(39,84)(40,83)(41,87)(42,86)(43,90)(44,89)(45,88);;
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*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(90)!( 6,36)( 7,37)( 8,38)( 9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(51,81)(52,82)(53,83)(54,84)(55,85)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80);
s1 := Sym(90)!(16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,86)(62,87)(63,88)(64,89)(65,90)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)(73,83)(74,84)(75,85);
s2 := Sym(90)!( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,16)( 7,20)( 8,19)( 9,18)(10,17)(11,26)(12,30)(13,29)(14,28)(15,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(46,66)(47,70)(48,69)(49,68)(50,67)(51,61)(52,65)(53,64)(54,63)(55,62)(56,71)(57,75)(58,74)(59,73)(60,72)(76,81)(77,85)(78,84)(79,83)(80,82)(87,90)(88,89);
s3 := Sym(90)!( 1,47)( 2,46)( 3,50)( 4,49)( 5,48)( 6,52)( 7,51)( 8,55)( 9,54)(10,53)(11,57)(12,56)(13,60)(14,59)(15,58)(16,62)(17,61)(18,65)(19,64)(20,63)(21,67)(22,66)(23,70)(24,69)(25,68)(26,72)(27,71)(28,75)(29,74)(30,73)(31,77)(32,76)(33,80)(34,79)(35,78)(36,82)(37,81)(38,85)(39,84)(40,83)(41,87)(42,86)(43,90)(44,89)(45,88);
poly := sub<Sym(90)|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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.