Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #2273⋊C3 |
| Fitting, derived, Sylow | #2273 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | #2273 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2273 | |
| UCS | C1 ◁ C2 | |
187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,5)(6,7,8)(9,10,13)(11,14,19)(12,16,18)(15,20,17);(2,6)(3,7);(1,5)(2,3)(4,8)(6,7)(9,17,12,11)(13,19,18,20)Polycyclic presentation:
3,2^7|b^2=f,c^2=fg,b^a=cf,c^a=bc,d^a=e,e^a=de,f^a=fg,g^a=f,h^a=fh,e^b=efgh,d^c=dfgh28 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5861);