Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C8×Q8⋊C9 |
| unambiguous semidirect product | (C8×Q8)⋊C9 |
| derived | Q8 ⋊ C8×C9 |
| Sylow | C8×Q8 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C8 | A4 |
| commutator, hypocenter | Q8 | C8×C9 |
| Frattini | C2×C3×C4 | C2×A4 |
| Fitting | C3×C8×Q8 | C3 |
| socle, prime radical | C22×C3 | C4×A4 |
| cosocle | C3×C4×Q8 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3×C8 | |
69 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C16×Q8⋊C9, C2×C8×Q8⋊C9, M16×Q8⋊C9, … |
| maximal normal subgroup of | C16×Q8⋊C9, C2×C8×Q8⋊C9, M16×Q8⋊C9, … |
| maximal quotient of | C16×Q8⋊C9, C2×C8×Q8⋊C9, C8×Q8⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 203);