Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C22 |
| commutator | C2 | C2×C3×C4×C8 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
384 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(9,10,11,13,12,14,15,16)(17,18,19);(1,2,3,4)(5,6)(7,8);(6,8)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5042);