Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3,4}

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

Overview

Group
SmallGroup(1152,155790)
Rank
4
Schläfli Type
{6,3,4}
Vertices, edges, …
24, 72, 48, 8
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Locally Toroidal
  • 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

24 vertex figures

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

8 facets

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

12 vertex figures

P/N, where N=<(s2*s3)^2, (s0*s1)^3> of order 4

4 facets

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

12 vertex figures

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

4 facets

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

12 vertex figures

  • 12 of 2-fold non-regular quotient of {3,4}*48

Representations

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

References

None.

to this polytope.