Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | D3×GU(2,𝔽3) |
| derived | C3×SL(2,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×S4 |
| commutator, hypocenter | C3×SL(2,𝔽3) | C2×C4 |
| Fitting | C3×#41 | C2×D3 |
| socle | C2×C3 | C2×SL(2,ℤ/4) |
| prime radical | D3×#41⋊C3 | C2 |
| cosocle | C3×#41⋊C3 | C22 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
169 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 169 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#5404, D3×#5405, #11921832, … |
| maximal normal subgroup of | D3×#5404, D3×#5405, #11921832, … |
| maximal quotient of | #11920497, #11920500, #11920542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5109);