Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #338⋊C9 |
| derived, Sylow | #338 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | C42⋊C3 |
| commutator, hypocenter | #338 | C9 |
| Frattini | C22×C3×C4 | A4 |
| Fitting, cosocle | C3×#338 | C3 |
| socle | C2×C3 | #95⋊C3 |
| prime radical | C23×C3 | SL(2,𝔽3) |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C3×C4 | |
45 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
3^2,2^6|a^3=b,c^2=ef,d^2=e,e^2=h,g^2=h,c^a=d,d^a=cdf,e^a=fgh,f^a=efg,d^c=dg,e^c=eh,f^c=fh,f^d=fh54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4);