Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 132120576 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C29×C4 |
| abelian | C23×C3×C42 |
| multiplicative | (ℤ/1560)ˣ, (ℤ/1680)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22 | C25×C3 |
| socle, prime radical | C25×C3 | C22 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
3000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 17 points:
(1,2,3,4)(15,16,17);(5,6,7,8);(9,10);(11,12);(13,14)384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19878);