Polytope of Type {6,4} with group SmallGroup(1920,240864)

Overview

Group: SmallGroup(1920,240864)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 240, 480, 160
Order of s₀s₁s₂: 40
Order of s₀s₁s₂s₁: 12
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{6,4}*960

4-fold

{6,4}*480

8-fold

16-fold

{6,4}*120

120-fold

{2,4}*16

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

80 facets

80 of {6}*12

124 vertex figures

8 of {2}*4
116 of {4}*8

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

82 facets

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

120 vertex figures

120 of {4}*8

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

64 facets

48 of {6}*12
16 of {2}*4

80 vertex figures

80 of {4}*8

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

40 facets

40 of {6}*12

62 vertex figures

4 of {2}*4
58 of {4}*8

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

42 facets

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

62 vertex figures

4 of {2}*4
58 of {4}*8

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1*s2> of order 5

32 facets

32 of {6}*12

48 vertex figures

48 of {4}*8

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

32 facets

24 of {6}*12
8 of {2}*4

44 vertex figures

8 of {2}*4
36 of {4}*8

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

34 facets

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

40 vertex figures

40 of {4}*8

P/N, where N=<(s1*s2)^2, s0*s1*s0*s2*s1*s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1*s2> of order 10

16 facets

16 of {6}*12

28 vertex figures

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

P/N, where N=<(s1*s2)^2, (s0*s1)^2*(s2*s1*s0)^2*s1*s2> of order 20

8 facets

8 of {6}*12

14 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := ( 1,19)( 2,26)( 3,10)( 4,33)( 5,34)( 6,35)( 7,38)( 8,16)( 9,42)(11,20)(12,21)(13,22)(14,48)(15,24)(17,27)(18,29)(23,36)(25,40)(28,32)(30,45)(31,46)(37,44)(39,41)(43,47)(49,84)(50,70)(51,78)(52,72)(53,64)(54,68)(55,75)(56,66)(57,58)(59,79)(60,89)(61,95)(62,69)(63,73)(65,80)(67,94)(71,85)(74,93)(76,82)(77,86)(81,87)(83,92)(88,96)(90,91);;
s1 := ( 1,58)( 2,64)( 3,67)( 4,68)( 5,69)( 6,70)( 7,72)( 8,74)( 9,75)(10,89)(11,81)(12,82)(13,83)(14,84)(15,86)(16,79)(17,90)(18,62)(19,87)(20,80)(21,95)(22,73)(23,96)(24,78)(25,54)(26,94)(27,85)(28,59)(29,71)(30,55)(31,50)(32,52)(33,76)(34,91)(35,88)(36,66)(37,57)(38,93)(39,51)(40,61)(41,49)(42,92)(43,60)(44,65)(45,63)(46,56)(47,53)(48,77);;
s2 := ( 1,58)( 2,64)( 3,86)( 4,68)( 5,69)( 6,70)( 7,72)( 8,96)( 9,75)(10,77)(11,74)(12,67)(13,82)(14,84)(15,83)(16,88)(17,81)(18,63)(19,57)(20,93)(21,94)(22,76)(23,90)(24,92)(25,71)(26,53)(27,87)(28,89)(29,73)(30,56)(31,51)(32,60)(33,54)(34,62)(35,50)(36,91)(37,61)(38,52)(39,79)(40,85)(41,59)(42,55)(43,80)(44,95)(45,66)(46,78)(47,65)(48,49);;
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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(96)!( 1,19)( 2,26)( 3,10)( 4,33)( 5,34)( 6,35)( 7,38)( 8,16)( 9,42)(11,20)(12,21)(13,22)(14,48)(15,24)(17,27)(18,29)(23,36)(25,40)(28,32)(30,45)(31,46)(37,44)(39,41)(43,47)(49,84)(50,70)(51,78)(52,72)(53,64)(54,68)(55,75)(56,66)(57,58)(59,79)(60,89)(61,95)(62,69)(63,73)(65,80)(67,94)(71,85)(74,93)(76,82)(77,86)(81,87)(83,92)(88,96)(90,91);
s1 := Sym(96)!( 1,58)( 2,64)( 3,67)( 4,68)( 5,69)( 6,70)( 7,72)( 8,74)( 9,75)(10,89)(11,81)(12,82)(13,83)(14,84)(15,86)(16,79)(17,90)(18,62)(19,87)(20,80)(21,95)(22,73)(23,96)(24,78)(25,54)(26,94)(27,85)(28,59)(29,71)(30,55)(31,50)(32,52)(33,76)(34,91)(35,88)(36,66)(37,57)(38,93)(39,51)(40,61)(41,49)(42,92)(43,60)(44,65)(45,63)(46,56)(47,53)(48,77);
s2 := Sym(96)!( 1,58)( 2,64)( 3,86)( 4,68)( 5,69)( 6,70)( 7,72)( 8,96)( 9,75)(10,77)(11,74)(12,67)(13,82)(14,84)(15,83)(16,88)(17,81)(18,63)(19,57)(20,93)(21,94)(22,76)(23,90)(24,92)(25,71)(26,53)(27,87)(28,89)(29,73)(30,56)(31,51)(32,60)(33,54)(34,62)(35,50)(36,91)(37,61)(38,52)(39,79)(40,85)(41,59)(42,55)(43,80)(44,95)(45,66)(46,78)(47,65)(48,49);
poly := sub<Sym(96)|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*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.