Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,4,10}

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

Overview

Group
SmallGroup(1440,5890)
Rank
4
Schläfli Type
{6,4,10}
Vertices, edges, …
18, 36, 60, 10
Order of s0s1s2s3
20
Order of s0s1s2s3s2s1
2
Also known as
{{6,4}4,{4,10|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

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)^3> of order 2

10 facets

  • 10 of 2-fold non-regular quotient of {6,4}*144

9 vertex figures

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

10 facets

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

6 vertex figures

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

10 facets

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

6 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*(s1*s0)^2*s1*s2> of order 6

10 facets

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

3 vertex figures

Representations

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