Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1320 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C23 | C2×C3×C8 |
| Frattini, cosocle | C2×M16 | C22×C3 |
| socle | C2×C3 | #323 |
| prime radical | C24×C3 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
128 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16)(17,18,19);(2,6)(8,13)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 456);