Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #99 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C2×D4 | C23×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G | |
278 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(7,11,12,17)(8,14,15,21)(9,16)(13,19,20,18);(6,7,10,13)(8,11,15,19)(9,12,16,20)(14,18,21,17)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5410);