Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9,6}

Atlas Canonical Name {6,9,6}*1944d

Overview

Group
SmallGroup(1944,2344)
Rank
4
Schläfli Type
{6,9,6}
Vertices, edges, …
6, 81, 81, 18
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

81-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=<s1*s2*s3*s2*s1*(s3*s2)^2*s3> of order 3

6 facets

6 vertex figures

P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2> of order 3

6 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,46)(11,48)(12,47)(13,49)(14,51)(15,50)(16,52)(17,54)(18,53)(19,37)(20,39)(21,38)(22,40)(23,42)(24,41)(25,43)(26,45)(27,44)(56,57)(59,60)(62,63)(64,73)(65,75)(66,74)(67,76)(68,78)(69,77)(70,79)(71,81)(72,80);;
s2 := ( 1,13)( 2,15)( 3,14)( 4,17)( 5,16)( 6,18)( 7,12)( 8,11)( 9,10)(19,20)(22,24)(26,27)(28,67)(29,69)(30,68)(31,71)(32,70)(33,72)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,55)(41,57)(42,56)(43,59)(44,58)(45,60)(46,74)(47,73)(48,75)(49,78)(50,77)(51,76)(52,79)(53,81)(54,80);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)(40,52)(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);;
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*s0*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)(10,46)(11,48)(12,47)(13,49)(14,51)(15,50)(16,52)(17,54)(18,53)(19,37)(20,39)(21,38)(22,40)(23,42)(24,41)(25,43)(26,45)(27,44)(56,57)(59,60)(62,63)(64,73)(65,75)(66,74)(67,76)(68,78)(69,77)(70,79)(71,81)(72,80);
s2 := Sym(81)!( 1,13)( 2,15)( 3,14)( 4,17)( 5,16)( 6,18)( 7,12)( 8,11)( 9,10)(19,20)(22,24)(26,27)(28,67)(29,69)(30,68)(31,71)(32,70)(33,72)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,55)(41,57)(42,56)(43,59)(44,58)(45,60)(46,74)(47,73)(48,75)(49,78)(50,77)(51,76)(52,79)(53,81)(54,80);
s3 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)(40,52)(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);
poly := sub<Sym(81)|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*s0*s1*s0*s1*s2*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.