Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×#5948 |
| Fitting | C22×C3×#41 ⋊ C3 |
| Sylow | C22×#41 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C2×C32 |
| Frattini | C2 | C2×C3×C22⋊A4 |
| Fitting | C22×C3×#41 | C3 |
| socle | C23×C3 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C3×C4 | |
382 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#1780⋊C3, C3×#2388⋊C3, C3×#2615⋊C3, … |
| maximal normal subgroup of | C3×#1780⋊C3, C3×#2388⋊C3, C3×#2615⋊C3, … |
| maximal quotient of | C3×C4×C22⋊SL(2,𝔽3), C3×#3459⋊C3, C3×#41⋊SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8602);