Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #346522 ⋊ C3 |
1266 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(3,8)(6,7)(9,10,13,16)(11,15,18,21,12,24,17,23,20,14,19,22);(1,2)(3,5)(4,7)(6,8)(9,18)(10,22)(11,12,17,19)(13,20)(14,15,21,23)(16,24)Polycyclic presentation:
2^2,3,2^7|b^2=j,d^2=e,f^2=j,g^2=j,b^a=bd,d^a=de,d^b=de,f^a=fh,f^c=fg,g^a=gi,g^c=f,g^f=gj,h^c=hij,h^g=hj,i^c=h,i^f=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72341937, #72342072, #72342078, … |
| maximal normal subgroup of | #72341937, #72342072, #72342078, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);