Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C3×#2988 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator | C2 | C23×C3×C8 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3,5,4,6,7,8)(10,13)(11,15)(17,18,19);(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(9,11,12,15)(10,14,13,16)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17428);