Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,3,6,10}

Atlas Canonical Name {4,3,6,10}*1440

Overview

Group
SmallGroup(1440,5871)
Rank
5
Schläfli Type
{4,3,6,10}
Vertices, edges, …
4, 6, 9, 30, 10
Order of s0s1s2s3s4
30
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

5-fold

6-fold

15-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP) 
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60);;
s1 := ( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(21,41)(22,42)(23,44)(24,43)(25,45)(26,46)(27,48)(28,47)(29,49)(30,50)(31,52)(32,51)(33,53)(34,54)(35,56)(36,55)(37,57)(38,58)(39,60)(40,59);;
s2 := ( 1,21)( 2,24)( 3,23)( 4,22)( 5,25)( 6,28)( 7,27)( 8,26)( 9,29)(10,32)(11,31)(12,30)(13,33)(14,36)(15,35)(16,34)(17,37)(18,40)(19,39)(20,38)(42,44)(46,48)(50,52)(54,56)(58,60);;
s3 := ( 5,17)( 6,18)( 7,19)( 8,20)( 9,13)(10,14)(11,15)(12,16)(21,41)(22,42)(23,43)(24,44)(25,57)(26,58)(27,59)(28,60)(29,53)(30,54)(31,55)(32,56)(33,49)(34,50)(35,51)(36,52)(37,45)(38,46)(39,47)(40,48);;
s4 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,17)(10,18)(11,19)(12,20)(21,25)(22,26)(23,27)(24,28)(29,37)(30,38)(31,39)(32,40)(41,45)(42,46)(43,47)(44,48)(49,57)(50,58)(51,59)(52,60);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(60)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60);
s1 := Sym(60)!( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(21,41)(22,42)(23,44)(24,43)(25,45)(26,46)(27,48)(28,47)(29,49)(30,50)(31,52)(32,51)(33,53)(34,54)(35,56)(36,55)(37,57)(38,58)(39,60)(40,59);
s2 := Sym(60)!( 1,21)( 2,24)( 3,23)( 4,22)( 5,25)( 6,28)( 7,27)( 8,26)( 9,29)(10,32)(11,31)(12,30)(13,33)(14,36)(15,35)(16,34)(17,37)(18,40)(19,39)(20,38)(42,44)(46,48)(50,52)(54,56)(58,60);
s3 := Sym(60)!( 5,17)( 6,18)( 7,19)( 8,20)( 9,13)(10,14)(11,15)(12,16)(21,41)(22,42)(23,43)(24,44)(25,57)(26,58)(27,59)(28,60)(29,53)(30,54)(31,55)(32,56)(33,49)(34,50)(35,51)(36,52)(37,45)(38,46)(39,47)(40,48);
s4 := Sym(60)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,17)(10,18)(11,19)(12,20)(21,25)(22,26)(23,27)(24,28)(29,37)(30,38)(31,39)(32,40)(41,45)(42,46)(43,47)(44,48)(49,57)(50,58)(51,59)(52,60);
poly := sub<Sym(60)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

References

None.

to this polytope.