Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C32 · Schur multiplier (H2) = C2 |
| direct product | C3×C4⋊C32 |
| unambiguous semidirect product | C4⋊(C3×C32) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C16 | C22 |
| commutator | C2 | C2×C3×C32 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| socle, prime radical | C22×C3 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C16 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 39 points:
(2,4)(5,6,7,11,8,12,15,21,9,13,16,22,18,24,27,31,10,14,17,23,19,25,28,32,20,26,29,33,30,34,35,36)(37,38,39);(1,2,3,4)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 557);