Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
150 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24)(25,26,27);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5523);