Polytope of Type {3,6,15} with group SmallGroup(540,54)

Overview

Group: SmallGroup(540,54)
Rank: 4
Schläfli Type: {3,6,15}
Vertices, edges, …: 3, 9, 45, 15
Order of s₀s₁s₂s₃: 15
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{3,2,15}*180

5-fold

{3,6,3}*108

9-fold

{3,2,5}*60

15-fold

{3,2,3}*36

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, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45);;
s1 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,17)(19,20)(22,23)(25,26)(28,29)(31,33)(34,36)(37,39)(40,42)(43,45);;
s2 := ( 1,16)( 2,18)( 3,17)( 4,28)( 5,30)( 6,29)( 7,25)( 8,27)( 9,26)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)(39,41);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 7,13)( 8,15)( 9,14)(11,12)(16,34)(17,36)(18,35)(19,31)(20,33)(21,32)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(28,37)(29,39)(30,38);;
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, s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(45)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45);
s1 := Sym(45)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,17)(19,20)(22,23)(25,26)(28,29)(31,33)(34,36)(37,39)(40,42)(43,45);
s2 := Sym(45)!( 1,16)( 2,18)( 3,17)( 4,28)( 5,30)( 6,29)( 7,25)( 8,27)( 9,26)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)(39,41);
s3 := Sym(45)!( 1, 4)( 2, 6)( 3, 5)( 7,13)( 8,15)( 9,14)(11,12)(16,34)(17,36)(18,35)(19,31)(20,33)(21,32)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(28,37)(29,39)(30,38);
poly := sub<Sym(45)|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, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References