Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | #2679⋊C3 |
| Fitting, derived, Sylow | #2679 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #2679 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C2 | |
215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,5)(3,6,11)(4,8,10)(7,12,9);(3,9)(6,7);(1,9,4,3)(5,11,10,12)Polycyclic presentation:
3,2^7|b^2=g,c^2=f,b^a=cg,c^a=bc,d^a=e,e^a=de,f^a=fg,g^a=f,h^a=fh,d^b=dh,e^b=egh,d^c=df,e^c=efh16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5863);