Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
274 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 19 points:
(2,7)(3,10)(4,5)(8,15)(9,13)(11,12)(17,18,19);(1,2)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15);(1,3,4,11,6,12,5,10)(2,8,7,14,9,15,13,16)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5339);