Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#578⋊C3 |
| unambiguous semidirect product | #578⋊C32 |
| unambiguous semidirect product | (C3×#578)⋊C3 |
| Fitting | C3×#578 ⋊ C3 |
| Sylow | #578 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D4×A4 |
| commutator, cosocle | #41 | C22×C32 |
| Frattini | C4 | C22×C3×A4 |
| Fitting | C3×#578 | C3 |
| hypercenter | C3×D8 | A4 |
| hypocenter | Q8 | C32×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8 | |
184 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#3416⋊C3, C3×#3417⋊C3, C2×C3×#578⋊C3, … |
| maximal normal subgroup of | C3×#3416⋊C3, C3×#3417⋊C3, C2×C3×#578⋊C3, … |
| maximal quotient of | C3×#2898⋊C3, C3×#2913⋊C3, C3×#3359⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7439);