Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#401 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,4,5);(8,12)(10,13)(15,18)(16,20)(22,23,25,26)(24,28,29,27);(6,7,9,11)(8,13,12,10)(14,15)(16,19)(17,18)(20,21)(23,26)(24,29);(1,2)(6,8)(7,10)(9,12)(11,13)(14,16,17,20)(15,19,18,21)(22,24,25,29)(23,27,26,28)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16840);