Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,6}

Atlas Canonical Name {4,6,6}*288f

Overview

Group
SmallGroup(288,1028)
Rank
4
Schläfli Type
{4,6,6}
Vertices, edges, …
4, 12, 18, 6
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.