Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C3×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C22 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C22 | C25×C3 |
| socle | C24×C3 | C23 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(5,6,7);(3,4);(8,9,10,11)(13,16)(17,19);(12,13)(14,17)(15,16)(18,19);(1,2)(3,4)(12,14,15,18)(13,17,16,19)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19881);