Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#5402 |
| Fitting | C22×C3×C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C22×C3 |
| Frattini | C4 | C2×C3×S4 |
| Fitting | C22×C3×C8 | D3 |
| socle | C23×C3 | D12 |
| hypercenter | C3×C8 | S4 |
| hypocenter | A4 | C3×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 | |
166 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 166 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.
| transitive degree | μt(G) = 72 |
66 conjugacy classes — view the full character table.
| maximal subgroup of | #11920579, #11920580, #11920585, … |
| maximal normal subgroup of | #11920579, #11920580, #11920585, … |
| maximal quotient of | C3×#73013, C3×#73022, C3×#73024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5470);