Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,4,6}

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

Overview

Group
SmallGroup(864,4686)
Rank
4
Schläfli Type
{6,4,6}
Vertices, edges, …
6, 36, 36, 18
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
{{6,4|2},{4,6}4}. 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=<s1*s2*s1*(s3*s2*s1*s2)^2*s3> of order 2

9 facets

6 vertex figures

  • 6 of 2-fold non-regular quotient of {4,6}*144
P/N, where N=<(s2*s1*s2*s3)^2> of order 3

6 facets

6 vertex figures

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

6 facets

6 vertex figures

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

3 facets

6 vertex figures

  • 6 of 6-fold non-regular quotient of {4,6}*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, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);;
s2 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);;
s3 := ( 1,40)( 2,41)( 3,42)( 4,37)( 5,38)( 6,39)( 7,43)( 8,44)( 9,45)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(16,34)(17,35)(18,36)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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)!( 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, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);
s2 := Sym(54)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54);
s3 := Sym(54)!( 1,40)( 2,41)( 3,42)( 4,37)( 5,38)( 6,39)( 7,43)( 8,44)( 9,45)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(16,34)(17,35)(18,36)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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.