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

Overview

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

{14,6}*168

3-fold

{28,2}*112

6-fold

{14,2}*56

7-fold

{4,6}*48a

12-fold

{7,2}*28

14-fold

{2,6}*24

21-fold

{4,2}*16

28-fold

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

Permutation Representation (Magma)

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

References