Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C3 |
| direct product | C3⋊D3×#125 |
| Fitting | C32×#125 ⋊ C2 |
| Sylow | C32 ⋊ C2×#125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#125 | C2 |
| prime radical | C22×C3⋊D3 | C23 |
| hypercenter | #125 | C3⋊D3 |
| hypocenter | C32 | C2×#125 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #125 | |
678 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#330, #11775686, #11775690, … |
| maximal normal subgroup of | C3⋊D3×#330, #11775686, #11775690, … |
| maximal quotient of | C3⋊D3×#382, C3⋊D3×#387, C3⋊D3×#389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8040);