Polytope of Type {6,28} with group SmallGroup(336,149)

Overview

Group: SmallGroup(336,149)
Rank: 3
Schläfli Type: {6,28}
Vertices, edges, …: 6, 84, 28
Order of s₀s₁s₂: 84
Order of s₀s₁s₂s₁: 2
Also known as: {6,28|2}. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

2-fold

{6,14}*168

3-fold

{2,28}*112

6-fold

{2,14}*56

7-fold

{6,4}*48a

12-fold

{2,7}*28

14-fold

{6,2}*24

21-fold

{2,4}*16

28-fold

{3,2}*12

42-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References