Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×D3×#120 |
| Fitting | C32×#120 ⋊ C2 |
| Sylow | C32 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C3 |
| Frattini | C22 | C23×C3×D3 |
| Fitting | C32×#120 | C2 |
| socle | C22×C32 | C24 |
| hypercenter | C3×#120 | D3 |
| hypocenter | C3 | C2×C3×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#120 | |
1389 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 1389 conjugacy classes of subgroups — too many to draw. Only its 262 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×AΣL(1,𝔽16), D3×#4536, C3×D3×B(4,𝔽2), … |
| maximal normal subgroup of | C3×D3×AΣL(1,𝔽16), D3×#4536, C3×D3×B(4,𝔽2), … |
| maximal quotient of | C3×D3×#379, C3×D3×#385, C3×D3×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7567);