Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×D3×C16 |
| Fitting | C3×C4×C16 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C16 | D3 |
| commutator, hypocenter | C3 | C2×C4×C16 |
| Frattini | C2×C8 | C22×D3 |
| Fitting | C3×C4×C16 | C2 |
| socle | C22×C3 | C22×C8 |
| prime radical | C22×D3 | C2×C8 |
| cosocle | C2×C3×C8 | C23 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C16 | |
216 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3);(1,2,3)(8,9,10,13,11,14,16,19,12,15,17,20,18,21,22,23);(4,5,6,7)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1625);