Polytope of Type {4,30} with group SmallGroup(720,784)

Overview

Group: SmallGroup(720,784)
Rank: 3
Schläfli Type: {4,30}
Vertices, edges, …: 12, 180, 90
Order of s₀s₁s₂: 20
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

5-fold

{4,6}*144

9-fold

{4,10}*80

10-fold

{4,6}*72

18-fold

{2,10}*40

36-fold

{2,5}*20

45-fold

{4,2}*16

90-fold

{2,2}*8

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=<s0*(s1*s2)^2*s1*s0*(s2*s1)^3> of order 2

45 facets

45 of {4}*8

6 vertex figures

6 of {30}*60

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

45 facets

45 of {4}*8

7 vertex figures

5 of {30}*60
2 of {15}*30

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

30 facets

30 of {4}*8

4 vertex figures

4 of {30}*60

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

30 facets

30 of {4}*8

8 vertex figures

2 of {30}*60
6 of {10}*20

P/N, where N=<s0*(s1*s2)^2*s1*s0*(s2*s1)^2*s2, s0*(s2*s1)^3*s0*(s2*s1)^2> of order 6

15 facets

15 of {4}*8

5 vertex figures

3 of {10}*20
2 of {15}*30

Representations

Permutation Representation (GAP)

s0 := ( 6,36)( 7,37)( 8,38)( 9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35);;
s1 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(16,41)(17,45)(18,44)(19,43)(20,42)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37);;
s2 := ( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)(11,27)(12,26)(13,30)(14,29)(15,28)(31,37)(32,36)(33,40)(34,39)(35,38)(41,42)(43,45);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

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

References

None.

to this polytope.