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

Overview

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

{10,2}*40

36-fold

{5,2}*20

90-fold

{2,2}*8

Covers minimal covers in bold

2-fold

{20,4}*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)^3> of order 2

9 facets

9 of {20}*40

45 vertex figures

45 of {4}*8

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

9 facets

9 of {20}*40

45 vertex figures

45 of {4}*8

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

10 facets

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

50 vertex figures

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

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

6 facets

6 of {20}*40

30 vertex figures

30 of {4}*8

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

5 facets

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

25 vertex figures

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

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

4 facets

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

20 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.