Polytope of Type {4,20} with group SmallGroup(720,784)

Overview

Group: SmallGroup(720,784)
Rank: 3
Schläfli Type: {4,20}
Vertices, edges, …: 18, 180, 90
Order of s₀s₁s₂: 30
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

{4,4}*144

10-fold

{4,4}*72

18-fold

{2,10}*40

36-fold

{2,5}*20

90-fold

{2,2}*8

Covers minimal covers in bold

2-fold

{4,20}*1440

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

45 facets

45 of {4}*8

9 vertex figures

9 of {20}*40

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

45 facets

45 of {4}*8

9 vertex figures

9 of {20}*40

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

50 facets

10 of {2}*4
40 of {4}*8

10 vertex figures

8 of {20}*40
2 of {10}*20

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

30 facets

30 of {4}*8

6 vertex figures

6 of {20}*40

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

25 facets

5 of {2}*4
20 of {4}*8

5 vertex figures

4 of {20}*40
1 of {10}*20

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

20 facets

10 of {2}*4
10 of {4}*8

4 vertex figures

2 of {20}*40
2 of {10}*20

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)( 6,32)( 7,31)( 8,35)( 9,34)(10,33)(11,12)(13,15)(16,37)(17,36)(18,40)(19,39)(20,38)(26,27)(28,30)(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*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0 ];;
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)( 6,32)( 7,31)( 8,35)( 9,34)(10,33)(11,12)(13,15)(16,37)(17,36)(18,40)(19,39)(20,38)(26,27)(28,30)(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*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0 >;

References

None.

to this polytope.