Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#1317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #99 |
| commutator | C23 | C2×C3×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C3 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C22⋊C4 ◁ G | |
150 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23)(25,26,27);(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 453);