Polytope of Type {8,4} with group SmallGroup(576,5357)

Overview

Group: SmallGroup(576,5357)
Rank: 3
Schläfli Type: {8,4}
Vertices, edges, …: 72, 144, 36
Order of s₀s₁s₂: 24
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

{4,4}*288

4-fold

{4,4}*144

8-fold

{4,4}*72

9-fold

{8,4}*64b

18-fold

{4,4}*32

36-fold

72-fold

{2,2}*8

Covers minimal covers in bold

2-fold

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

18 facets

18 of {8}*16

36 vertex figures

36 of {4}*8

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

12 facets

12 of {8}*16

24 vertex figures

24 of {4}*8

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

12 facets

12 of {8}*16

24 vertex figures

24 of {4}*8

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

6 facets

6 of {8}*16

12 vertex figures

12 of {4}*8

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.