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 | C2×C3×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5);(7,10)(11,13)(14,15,17,16,18,19,21,20);(6,7,9,10)(8,11,12,13);(1,2)(6,8,9,12)(7,11,10,13)(14,16,17,19,18,20,21,15)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17399);