Polytope of Type {6,4} with group SmallGroup(432,530)

Overview

Group: SmallGroup(432,530)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 54, 108, 36
Order of s₀s₁s₂: 12
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

2-fold

{6,4}*216

3-fold

{6,4}*144

6-fold

{6,4}*72

27-fold

{2,4}*16

54-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-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*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 2

19 facets

17 of {6}*12
2 of {3}*6

27 vertex figures

27 of {4}*8

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

18 facets

18 of {6}*12

30 vertex figures

24 of {4}*8
6 of {2}*4

P/N, where N=<s0*s2*s1*s0*s1*s2> of order 3

16 facets

10 of {6}*12
6 of {2}*4

18 vertex figures

18 of {4}*8

P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 3

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

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

9 facets

9 of {6}*12

15 vertex figures

12 of {4}*8
3 of {2}*4

P/N, where N=<(s0*s1)^3, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4

10 facets

2 of {3}*6
8 of {6}*12

15 vertex figures

12 of {4}*8
3 of {2}*4

P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2> of order 6

6 facets

6 of {6}*12

12 vertex figures

6 of {2}*4
6 of {4}*8

P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 6

8 facets

5 of {6}*12
3 of {2}*4

12 vertex figures

6 of {4}*8
6 of {2}*4

P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s1*s2> of order 6

9 facets

2 of {3}*6
3 of {2}*4
4 of {6}*12

9 vertex figures

9 of {4}*8

Representations

Permutation Representation (GAP)

s0 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)(20,21)(22,23)(25,27)(28,37)(29,39)(30,38)(31,41)(32,40)(33,42)(34,45)(35,44)(36,43)(47,48)(49,50)(52,54);;
s2 := ( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,15)(16,22)(17,24)(18,23)(25,27)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,42)(43,49)(44,51)(45,50)(52,54);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

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

Permutation Representation (Magma)

s0 := Sym(54)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);
s1 := Sym(54)!( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)(20,21)(22,23)(25,27)(28,37)(29,39)(30,38)(31,41)(32,40)(33,42)(34,45)(35,44)(36,43)(47,48)(49,50)(52,54);
s2 := Sym(54)!( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,15)(16,22)(17,24)(18,23)(25,27)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,42)(43,49)(44,51)(45,50)(52,54);
poly := sub<Sym(54)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

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

References

None.

to this polytope.