Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×C42×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42 | C22 |
| commutator | C2 | C22×C3×C42 |
| Frattini, cosocle | C23 | C24×C3 |
| socle | C23×C3 | C24 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42 ◁ G | |
924 conjugacy classes of subgroups; only the 576 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 576 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(4,5,6,7);(8,9,10,11);(12,13,14,15);(13,15)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16711);