Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C25×C4 |
| abelian | C22×C3×C4×C8 |
| multiplicative | (ℤ/1120)ˣ, (ℤ/1248)ˣ, (ℤ/1440)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle, prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3,4,5,6,7,8)(17,18,19);(9,10,11,12);(13,14);(15,16)384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17309);