Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×S4 |
| unambiguous semidirect product | A4⋊(C2×SL(2,𝔽3)) |
| unambiguous semidirect product | (Q8×S4)⋊C3 |
| Fitting | C22×Q8 ⋊ C3×D3 |
| derived | Q8×A4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×S4 |
| commutator, cosocle, hypocenter | Q8×A4 | C2×C3 |
| Fitting | C22×Q8 | C3×D3 |
| socle | C23 | D3×A4 |
| LCS | G ▷ Q8×A4 | |
| UCS | C1 ◁ C2 | |
133 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 133 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
35 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#226, #11923446, #11923448, … |
| maximal normal subgroup of | S4×#226, #11923446, #11923448, … |
| maximal quotient of | SL(2,𝔽3)×#226, SL(2,𝔽3)×GL(2,𝔽3), SL(2,𝔽3)×SL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8272);