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

Overview

Group: SmallGroup(540,54)
Rank: 4
Schläfli Type: {15,6,3}
Vertices, edges, …: 15, 45, 9, 3
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

{15,2,3}*180

5-fold

{3,6,3}*108

9-fold

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

Permutation Representation (Magma)

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

References