Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#1436 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C23 |
| commutator | C22 | C2×C3×C42 |
| Frattini, cosocle | C24 | C23×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(5,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,18,19);(1,2,3,4)(10,12);(5,7,8,6)(14,16)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4727);