Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6,4}

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

Overview

Group
SmallGroup(864,4686)
Rank
4
Schläfli Type
{6,6,4}
Vertices, edges, …
6, 54, 36, 12
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
{{6,6|2},{6,4}4}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

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=<(s1*s2*s3*s2)^2> of order 3

4 facets

6 vertex figures

  • 6 of 3-fold non-regular quotient of {6,4}*144
P/N, where N=<s1*s3*s2*s1*s2*s3> of order 3

8 facets

6 vertex figures

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

Representations

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

References

None.

to this polytope.