Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C22⋊(#95⋊C3) |
| Fitting, derived, Sylow | C22×#95 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C22×#95 | C3 |
| Frattini | C23 | C22⋊A4 |
| socle | C25 | A4 |
| LCS | G ▷ C22×#95 | |
| UCS | C1 ◁ C2 | |
345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4,3)(5,6,9)(7,12,16)(8,10,15)(11,14,13);(1,4)(2,3);(5,13)(6,11,10,12)(7,8)(9,15)Polycyclic presentation:
3,2^7|b^2=f,c^2=h,b^a=bc,c^a=bfgh,d^a=de,e^a=d,f^a=gh,g^a=fh,h^a=f,c^b=cfg32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5859);