Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×C4×SL(2,𝔽3) |
| Fitting, Sylow | C22×C4×Q8 ⋊ C3 |
| derived | Q8 ⋊ C22×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C22 | C23×A4 |
| Fitting | C22×C4×Q8 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C23×SL(2,𝔽3) | C2 |
| cosocle | C2×Q8 | C23×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C4 | |
519 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(5,6,7,8)(10,11,13)(14,16,15);(3,4)(9,13,12,15)(10,16,14,11);(1,2)112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18151);