Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,6}

Atlas Canonical Name {4,4,6}*576

Overview

Group
SmallGroup(576,8418)
Rank
4
Schläfli Type
{4,4,6}
Vertices, edges, …
4, 24, 36, 18
Order of s0s1s2s3
4
Order of s0s1s2s3s2s1
2
Also known as
{{4,4|2},{4,6}4}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

9-fold

18-fold

36-fold

Covers minimal covers in bold

2-fold

3-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

4 vertex figures

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

6 facets

4 vertex figures

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

6 facets

4 vertex figures

  • 4 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

4 vertex figures

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

Representations

Permutation Representation (GAP) 
s0 := ( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);;
s1 := ( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(19,28)(20,31)(21,34)(22,29)(23,32)(24,35)(25,30)(26,33)(27,36)(38,40)(39,43)(42,44)(47,49)(48,52)(51,53)(55,64)(56,67)(57,70)(58,65)(59,68)(60,71)(61,66)(62,69)(63,72);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)(67,70)(68,71)(69,72);;
s3 := ( 1,41)( 2,40)( 3,42)( 4,38)( 5,37)( 6,39)( 7,44)( 8,43)( 9,45)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)(17,52)(18,54)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(28,68)(29,67)(30,69)(31,65)(32,64)(33,66)(34,71)(35,70)(36,72);;
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*s0*s1, 
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(72)!( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);
s1 := Sym(72)!( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(19,28)(20,31)(21,34)(22,29)(23,32)(24,35)(25,30)(26,33)(27,36)(38,40)(39,43)(42,44)(47,49)(48,52)(51,53)(55,64)(56,67)(57,70)(58,65)(59,68)(60,71)(61,66)(62,69)(63,72);
s2 := Sym(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)(67,70)(68,71)(69,72);
s3 := Sym(72)!( 1,41)( 2,40)( 3,42)( 4,38)( 5,37)( 6,39)( 7,44)( 8,43)( 9,45)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)(17,52)(18,54)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(28,68)(29,67)(30,69)(31,65)(32,64)(33,66)(34,71)(35,70)(36,72);
poly := sub<Sym(72)|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*s0*s1, 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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.