Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,4}

Atlas Canonical Name {3,6,4}*1152a

Overview

Group
SmallGroup(1152,155790)
Rank
4
Schläfli Type
{3,6,4}
Vertices, edges, …
12, 72, 96, 8
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

4-fold

6-fold

12-fold

16-fold

24-fold

48-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=<(s2*s3)^2> of order 2

4 facets

12 vertex figures

  • 12 of 2-fold non-regular quotient of {6,4}*96
P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1> of order 2

8 facets

  • 8 of 2-fold non-regular quotient of {3,6}*144

8 vertex figures

P/N, where N=<(s2*s3)^2, s0*s1*s0*(s2*s1)^2*s0*s2*s1> of order 4

4 facets

  • 4 of 2-fold non-regular quotient of {3,6}*144

8 vertex figures

  • 4 of 2-fold non-regular quotient of {6,4}*96
  • 4 of 2-fold non-regular quotient of {3,4}*48
P/N, where N=<(s2*s3)^2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 4

4 facets

  • 4 of 2-fold non-regular quotient of {3,6}*144

6 vertex figures

  • 6 of 2-fold non-regular quotient of {6,4}*96

Representations

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

References

None.

to this polytope.