Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | A4×GL(2,𝔽3) |
| unambiguous semidirect product | (C22×GL(2,𝔽3))⋊C3 |
| Fitting | C22×Q8 ⋊ C3×D3 |
| derived | C22×SL(2,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×S4 |
| commutator, cosocle, hypocenter | C22×SL(2,𝔽3) | C2×C3 |
| Fitting | C22×Q8 | C3×D3 |
| socle | C23 | C3×S4 |
| LCS | G ▷ C22×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
132 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
32 conjugacy classes — view the full character table.
| maximal subgroup of | #11923443, #11923446, #11923447, … |
| maximal normal subgroup of | #11923443, #11923446, #11923447, … |
| maximal quotient of | #85430⋊C3, SL(2,𝔽3)×GL(2,𝔽3), A4×#859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8271);