Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| Fitting | C3×#14417 ⋊ C2 |
| Sylow | C3 ⋊ #574938 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C24×D3 |
| commutator | C23×C3 | C24×C4 |
| Fitting | C3×#14417 | C2 |
| socle, cosocle | C24×C3 | C25 |
| prime radical | C22×#5609 | C2 |
| hypercenter | #14417 | D3 |
| hypocenter | C3 | #574938 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #14417 | |
15148 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^9,3|a^2=f,b^2=g,c^2=fhi,e^2=i,b^a=bfg,c^b=cfgh,d^a=dhi,d^b=di,e^a=ehi,e^b=ehi,e^c=ehi,e^d=ei,j^c=j^2168 conjugacy classes — view the full character table.
| maximal subgroup of | #72255049, #72255559, #72270171, … |
| maximal normal subgroup of | #72255049, #72255559, #72270171, … |
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, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.6*F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);