Polytope of Type {3,12,3} with group SmallGroup(1728,46101)

Overview

Group: SmallGroup(1728,46101)
Rank: 4
Schläfli Type: {3,12,3}
Vertices, edges, …: 12, 144, 144, 12
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 12
Also known as: if this polytope has a name.

Special Properties

Orientable
Self-Dual

Quotients maximal quotients in bold

3-fold

{3,4,3}*576

16-fold

{3,6,3}*108

48-fold

{3,2,3}*36

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*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 2

8 facets

4 of 2-fold non-regular quotient of {3,12}*144
4 of {3,12}*144

8 vertex figures

4 of 2-fold non-regular quotient of {12,3}*144
4 of {12,3}*144

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 4

6 facets

4 of {3,6}*36
2 of {3,12}*144

6 vertex figures

6 of 2-fold non-regular quotient of {12,3}*144

P/N, where N=<(s1*s2)^2*s1*s3*s2*s1*s2*s3, s1*s2*s1*s3*(s2*s1)^2*s3*s2> of order 4

6 facets

6 of 2-fold non-regular quotient of {3,12}*144

6 vertex figures

4 of {6,3}*36
2 of {12,3}*144

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*s2*s3*s2*s1*s0*s2*s1*s3*s2> of order 8

4 facets

2 of {3,6}*36
2 of 2-fold non-regular quotient of {3,12}*144

4 vertex figures

2 of 2-fold non-regular quotient of {12,3}*144
2 of {6,3}*36

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.