Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#1404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C3×C4 |
| Frattini, cosocle | #124 | C22×C3 |
| prime radical | C3×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#124 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,2,5,10)(3,13,9,6)(4,14,7,12)(8,11,15,16)(17,18,19);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 540);