Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#3529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| socle | C22×C3 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,27);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(2,6)(3,7)(11,18)(14,17)(19,24)(21,23);(2,6)(5,8)(10,15)(11,18)(16,22)(19,24);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19989);