Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,10}

Atlas Canonical Name {3,6,10}*1080

Overview

Group
SmallGroup(1080,287)
Rank
4
Schläfli Type
{3,6,10}
Vertices, edges, …
9, 27, 90, 10
Order of s0s1s2s3
30
Order of s0s1s2s3s2s1
2
Also known as
{{3,6}6,{6,10|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

5-fold

9-fold

15-fold

18-fold

45-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)^2*s1> of order 3

10 facets

  • 10 of 3-fold non-regular quotient of {3,6}*108

5 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45);;
s1 := ( 1,17)( 2,18)( 3,16)( 4,20)( 5,21)( 6,19)( 7,23)( 8,24)( 9,22)(10,26)(11,27)(12,25)(13,29)(14,30)(15,28);;
s2 := ( 2, 3)( 4,13)( 5,15)( 6,14)( 7,10)( 8,12)( 9,11)(17,18)(19,28)(20,30)(21,29)(22,25)(23,27)(24,26)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)(39,41);;
s3 := ( 1, 4)( 2, 5)( 3, 6)( 7,13)( 8,14)( 9,15)(16,19)(17,20)(18,21)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36)(37,43)(38,44)(39,45);;
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*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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(45)!(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45);
s1 := Sym(45)!( 1,17)( 2,18)( 3,16)( 4,20)( 5,21)( 6,19)( 7,23)( 8,24)( 9,22)(10,26)(11,27)(12,25)(13,29)(14,30)(15,28);
s2 := Sym(45)!( 2, 3)( 4,13)( 5,15)( 6,14)( 7,10)( 8,12)( 9,11)(17,18)(19,28)(20,30)(21,29)(22,25)(23,27)(24,26)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)(39,41);
s3 := Sym(45)!( 1, 4)( 2, 5)( 3, 6)( 7,13)( 8,14)( 9,15)(16,19)(17,20)(18,21)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36)(37,43)(38,44)(39,45);
poly := sub<Sym(45)|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*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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.