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

Overview

Group: SmallGroup(1296,3531)
Rank: 4
Schläfli Type: {6,4,4}
Vertices, edges, …: 18, 81, 54, 9
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Orientable
Flat

Quotients maximal quotients in bold

9-fold

{2,4,4}*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*s1)^2> of order 3

9 facets

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

6 vertex figures

6 of {4,4}*72

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

9 facets

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

6 vertex figures

6 of {4,4}*72

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.