Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#2162 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | C22 |
| commutator | C2 | C22×C3×C16 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C23×C3 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ G | |
160 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 120 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)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24)(25,26,27);(2,5)(6,8);(1,3,4,7)(2,6,5,8)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5453);