Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#2965 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ G | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14,13,15,16,11)(17,18,19);(2,5)(6,8);(9,11,12,10,13,14,16,15);(1,3,4,7)(2,6,5,8)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17405);