Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×#997 |
| Fitting | C2×C3×H(5,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C2×C32 |
| Sylow | C2×H(5,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C32 |
| Frattini | C2 | C2×C3×C22⋊A4 |
| Fitting | C2×C3×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22×C3 | |
438 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
66 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#1790⋊C3, C2×C3×B(4,𝔽2)⋊C3, C2×C3×#458⋊C3, … |
| maximal normal subgroup of | C3×#1790⋊C3, C2×C3×B(4,𝔽2)⋊C3, C2×C3×#458⋊C3, … |
| maximal quotient of | C3×C4×#997, C2×C3×#543⋊C3, C3×#3459⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8603);