Polytope of Type {3,4,6,3} with group SmallGroup(864,4673)

Overview

Group: SmallGroup(864,4673)
Rank: 5
Schläfli Type: {3,4,6,3}
Vertices, edges, …: 6, 12, 24, 9, 3
Order of s₀s₁s₂s₃s₄: 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{3,4,2,3}*288

4-fold

{3,2,6,3}*216

6-fold

{3,4,2,3}*144

12-fold

{3,2,2,3}*72

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> of order 2

3 facets

3 of 2-fold non-regular quotient of {3,4,6}*288

4 vertex figures

2 of {2,6,3}*72
2 of {4,6,3}*144

Representations

Permutation Representation (GAP)

s0 := ( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35);;
s1 := ( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)(31,32)(35,36);;
s2 := ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)(13,29)(14,30)(15,31)(16,32)(17,25)(18,26)(19,27)(20,28)(21,33)(22,34)(23,35)(24,36);;
s4 := ( 1,13)( 2,14)( 3,15)( 4,16)( 5,21)( 6,22)( 7,23)( 8,24)( 9,17)(10,18)(11,19)(12,20)(29,33)(30,34)(31,35)(32,36);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s4*s2*s3*s2*s3*s4*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(36)!( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35);
s1 := Sym(36)!( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)(31,32)(35,36);
s2 := Sym(36)!( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33);
s3 := Sym(36)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)(13,29)(14,30)(15,31)(16,32)(17,25)(18,26)(19,27)(20,28)(21,33)(22,34)(23,35)(24,36);
s4 := Sym(36)!( 1,13)( 2,14)( 3,15)( 4,16)( 5,21)( 6,22)( 7,23)( 8,24)( 9,17)(10,18)(11,19)(12,20)(29,33)(30,34)(31,35)(32,36);
poly := sub<Sym(36)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s4*s2*s3*s2*s3*s4*s2*s3 >;

References

None.

to this polytope.