Polytope of Type {4,6,6} with group SmallGroup(1296,3531)

Overview

Group: SmallGroup(1296,3531)
Rank: 4
Schläfli Type: {4,6,6}
Vertices, edges, …: 6, 54, 81, 18
Order of s₀s₁s₂s₃: 4
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Non-Orientable
Flat

Quotients maximal quotients in bold

9-fold

{4,6,2}*144

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=<s0*s1*s2*s3*s2*s1*s0*s2*s3*s2> of order 3

6 facets

6 of {4,6}*72

6 vertex figures

3 of {6,6}*72c
3 of {6,6}*72a

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

6 facets

6 of {4,6}*72

6 vertex figures

6 of 3-fold non-regular quotient of {6,6}*216d

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.