Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3162 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C22×D4 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,15)(13,17)(14,20)(16,21)(19,23)(22,24)(25,26,27);(1,3,4,8)(2,5,6,7)(11,19)(14,22)(18,23)(20,24);(9,11,12,20,13,19,21,24)(10,14,16,18,17,22,15,23);(9,12,13,21)(10,15,17,16)(11,20,19,24)(14,23,22,18)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17602);