Polytope of Type {6,6,3} with group SmallGroup(864,4673)

Overview

Group: SmallGroup(864,4673)
Rank: 4
Schläfli Type: {6,6,3}
Vertices, edges, …: 6, 72, 36, 12
Order of s₀s₁s₂s₃: 12
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

4-fold

{6,6,3}*216b

9-fold

{2,6,3}*96

12-fold

18-fold

{2,3,3}*48

24-fold

{3,2,3}*36

36-fold

{2,2,3}*24

Covers minimal covers in bold

2-fold

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

6 facets

6 of {6,6}*72a

6 vertex figures

6 of 2-fold non-regular quotient of {6,3}*144

P/N, where N=<(s1*s2)^2> of order 3

6 facets

3 of {6,2}*24
3 of {6,6}*72a

6 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := ( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)(31,35)(32,36);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,17)(14,19)(15,18)(16,20)(22,23)(25,29)(26,31)(27,30)(28,32)(34,35);;
s2 := ( 3, 4)( 7, 8)(11,12)(13,25)(14,26)(15,28)(16,27)(17,29)(18,30)(19,32)(20,31)(21,33)(22,34)(23,36)(24,35);;
s3 := ( 1,16)( 2,14)( 3,15)( 4,13)( 5,20)( 6,18)( 7,19)( 8,17)( 9,24)(10,22)(11,23)(12,21)(25,28)(29,32)(33,36);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(36)!( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)(31,35)(32,36);
s1 := Sym(36)!( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,17)(14,19)(15,18)(16,20)(22,23)(25,29)(26,31)(27,30)(28,32)(34,35);
s2 := Sym(36)!( 3, 4)( 7, 8)(11,12)(13,25)(14,26)(15,28)(16,27)(17,29)(18,30)(19,32)(20,31)(21,33)(22,34)(23,36)(24,35);
s3 := Sym(36)!( 1,16)( 2,14)( 3,15)( 4,13)( 5,20)( 6,18)( 7,19)( 8,17)( 9,24)(10,22)(11,23)(12,21)(25,28)(29,32)(33,36);
poly := sub<Sym(36)|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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 >;

References

None.

to this polytope.