Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×#131⋊C9 |
| unambiguous semidirect product | (C2×#131)⋊C9 |
| Sylow | C2×#131 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C8 | A4 |
| commutator, hypocenter | Q8 | C2×C4×C9 |
| Frattini | C3×C4 | C22×A4 |
| Fitting | C2×C3×#131 | C3 |
| socle | C22×C3 | C4×A4 |
| prime radical | C2×C3×#41 | C2×C3 |
| cosocle | C3×#41 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3×C8 | |
133 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 133 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#131⋊C9, #2875⋊C9, #2880⋊C9, … |
| maximal normal subgroup of | C4×#131⋊C9, #2875⋊C9, #2880⋊C9, … |
| maximal quotient of | C4×#131⋊C9, #2880⋊C9, C4⋊(#131⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1437);