Polytope of Type {20,6} with group SmallGroup(1920,238598)

Overview

Group: SmallGroup(1920,238598)
Rank: 3
Schläfli Type: {20,6}
Vertices, edges, …: 160, 480, 48
Order of s₀s₁s₂: 30
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

4-fold

{20,6}*480c

5-fold

{4,6}*384a

8-fold

{20,6}*240b

10-fold

{4,6}*192a

16-fold

{10,6}*120

20-fold

{4,6}*96

40-fold

48-fold

{10,2}*40

80-fold

96-fold

{5,2}*20

160-fold

{2,3}*12

240-fold

{2,2}*8

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*s1)^2*s2> of order 2

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

28 facets

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

80 vertex figures

80 of {6}*12

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

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

24 facets

24 of {20}*40

80 vertex figures

80 of {6}*12

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

18 facets

6 of {20}*40
12 of {10}*20

40 vertex figures

40 of {6}*12

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

16 facets

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

40 vertex figures

40 of {6}*12

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

16 facets

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

40 vertex figures

40 of {6}*12

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

14 facets

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

40 vertex figures

40 of {6}*12

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

14 facets

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

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

14 facets

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

40 vertex figures

40 of {6}*12

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

14 facets

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

40 vertex figures

40 of {6}*12

P/N, where N=<(s0*s1)^3*(s2*s1)^2*(s0*s1)^2*s2, s0*s1*s0*s2*(s1*s0)^2*(s1*s2)^2*s1*s0*s1> of order 4

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

14 facets

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

40 vertex figures

40 of {6}*12

P/N, where N=<(s0*s1)^3*(s2*s1)^2*s0*s1*s0*s2*s1*s2, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2> of order 4

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

12 facets

12 of {20}*40

40 vertex figures

40 of {6}*12

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

14 facets

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

40 vertex figures

40 of {6}*12

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

8 facets

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

20 vertex figures

20 of {6}*12

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

8 facets

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

20 vertex figures

20 of {6}*12

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

10 facets

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

20 vertex figures

20 of {6}*12

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

6 facets

6 of {20}*40

20 vertex figures

20 of {6}*12

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

6 facets

6 of {20}*40

20 vertex figures

20 of {6}*12

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

9 facets

3 of {20}*40
6 of {10}*20

20 vertex figures

20 of {6}*12

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

8 facets

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

20 vertex figures

20 of {6}*12

P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2, (s0*s1)^3*(s2*s1)^2*s0*s1*s0*s2*s1*s2, (s0*s1)^2*(s0*s2*s1)^2*s0*(s1*s2)^2*s1> of order 8

8 facets

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

20 vertex figures

20 of {6}*12

P/N, where N=<(s0*s1)^2*s2*s1*s0*s2*s1*s2, (s0*s1)^3*(s2*s1)^2*(s0*s1)^2*s2> of order 8

6 facets

6 of {20}*40

20 vertex figures

20 of {6}*12

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

7 facets

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

20 vertex figures

20 of {6}*12

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

6 facets

6 of {20}*40

20 vertex figures

20 of {6}*12

Representations

Permutation Representation (GAP)

s0 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56);;
s1 := ( 1,17)( 2,18)( 3,20)( 4,19)( 5,22)( 6,21)( 7,23)( 8,24)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,26)(16,25)(33,65)(34,66)(35,68)(36,67)(37,70)(38,69)(39,71)(40,72)(41,80)(42,79)(43,77)(44,78)(45,75)(46,76)(47,74)(48,73)(51,52)(53,54)(57,64)(58,63)(59,61)(60,62);;
s2 := ( 2, 4)( 5,13)( 6,16)( 7,15)( 8,14)(10,12)(18,20)(21,29)(22,32)(23,31)(24,30)(26,28)(34,36)(37,45)(38,48)(39,47)(40,46)(42,44)(50,52)(53,61)(54,64)(55,63)(56,62)(58,60)(66,68)(69,77)(70,80)(71,79)(72,78)(74,76);;
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*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, 
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, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(80)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56);
s1 := Sym(80)!( 1,17)( 2,18)( 3,20)( 4,19)( 5,22)( 6,21)( 7,23)( 8,24)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,26)(16,25)(33,65)(34,66)(35,68)(36,67)(37,70)(38,69)(39,71)(40,72)(41,80)(42,79)(43,77)(44,78)(45,75)(46,76)(47,74)(48,73)(51,52)(53,54)(57,64)(58,63)(59,61)(60,62);
s2 := Sym(80)!( 2, 4)( 5,13)( 6,16)( 7,15)( 8,14)(10,12)(18,20)(21,29)(22,32)(23,31)(24,30)(26,28)(34,36)(37,45)(38,48)(39,47)(40,46)(42,44)(50,52)(53,61)(54,64)(55,63)(56,62)(58,60)(66,68)(69,77)(70,80)(71,79)(72,78)(74,76);
poly := sub<Sym(80)|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*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, 
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, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1 >;

References

None.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle