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) = C210 |
| direct product | C23×C3×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | C22 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ G | |
2488 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)(7,8,9);(3,4);(5,6);(11,13)(14,15,16,17);(10,11,12,13)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19880);