Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#5923 |
| Fitting | C2×C3×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C3 |
| Frattini | C2 | C22×C3×S4 |
| Fitting | C2×C3×#41 | D3 |
| socle | C2×C3 | C22×S4 |
| hypercenter | C2×C3×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 | |
301 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 54 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 | #11920684, #11920715, C3×#73073, … |
| maximal normal subgroup of | #11920684, #11920715, C3×#73073, … |
| maximal quotient of | C3×#85318, C3×#85322, C3×#85325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8447);