Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,4,6}

Atlas Canonical Name {6,4,6}*864b

Overview

Group
SmallGroup(864,4686)
Rank
4
Schläfli Type
{6,4,6}
Vertices, edges, …
18, 36, 36, 6
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
{{6,4}4,{4,6|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

18-fold

27-fold

36-fold

54-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

6 facets

  • 6 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

6 facets

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

6 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2> of order 3

6 facets

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

6 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s1*s2> of order 6

6 facets

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

3 vertex figures

Representations

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

References

None.

to this polytope.