Polytope of Type {4,6,6} with group SmallGroup(1296,3490)

Overview

Group: SmallGroup(1296,3490)
Rank: 4
Schläfli Type: {4,6,6}
Vertices, edges, …: 4, 54, 81, 27
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Non-Orientable
Flat

Quotients maximal quotients in bold

No regular quotients.

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*s1*s3)^2*s2*s3> of order 3

9 facets

9 of {4,6}*48b

4 vertex figures

1 of {6,6}*108
3 of 3-fold non-regular quotient of {6,6}*324b

Representations

Permutation Representation (GAP)

s0 := ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12);;
s1 := ( 7,10)( 8,11)( 9,12);;
s2 := ( 2, 3)( 4,10)( 5,11)( 6,12)( 8, 9);;
s3 := ( 1, 2)( 4, 5)( 7, 8)(10,11);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s0*s2*s3*s2*s3*s2*s1*s0*s1*s2*s3*s2*s3*s2*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(12)!( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12);
s1 := Sym(12)!( 7,10)( 8,11)( 9,12);
s2 := Sym(12)!( 2, 3)( 4,10)( 5,11)( 6,12)( 8, 9);
s3 := Sym(12)!( 1, 2)( 4, 5)( 7, 8)(10,11);
poly := sub<Sym(12)|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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s0*s2*s3*s2*s3*s2*s1*s0*s1*s2*s3*s2*s3*s2*s1*s0 >;

References

None.

to this polytope.