Polytope of Type {14,6,3} with group SmallGroup(504,172)

Overview

Group: SmallGroup(504,172)
Rank: 4
Schläfli Type: {14,6,3}
Vertices, edges, …: 14, 42, 9, 3
Order of s₀s₁s₂s₃: 42
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{14,2,3}*168

6-fold

{7,2,3}*84

7-fold

{2,6,3}*72

21-fold

{2,2,3}*24

Covers minimal covers in bold

2-fold

3-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)(44,49)(45,48)(46,47)(51,56)(52,55)(53,54)(58,63)(59,62)(60,61);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 8,16)( 9,15)(10,21)(11,20)(12,19)(13,18)(14,17)(22,23)(24,28)(25,27)(29,37)(30,36)(31,42)(32,41)(33,40)(34,39)(35,38)(43,44)(45,49)(46,48)(50,58)(51,57)(52,63)(53,62)(54,61)(55,60)(56,59);;
s2 := ( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63);;
s3 := ( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,28)( 8,36)( 9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(63)!( 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)(44,49)(45,48)(46,47)(51,56)(52,55)(53,54)(58,63)(59,62)(60,61);
s1 := Sym(63)!( 1, 2)( 3, 7)( 4, 6)( 8,16)( 9,15)(10,21)(11,20)(12,19)(13,18)(14,17)(22,23)(24,28)(25,27)(29,37)(30,36)(31,42)(32,41)(33,40)(34,39)(35,38)(43,44)(45,49)(46,48)(50,58)(51,57)(52,63)(53,62)(54,61)(55,60)(56,59);
s2 := Sym(63)!( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63);
s3 := Sym(63)!( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,28)( 8,36)( 9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63);
poly := sub<Sym(63)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*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 >;

References