Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1327 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C2×C4 | C2×C3×C8 |
| Frattini, cosocle | C2×M16 | C22×C3 |
| socle | C2×C3 | #324 |
| prime radical | C23×C3 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,8,3,6,10,14,5,9,12,16,11,7,13,15)(17,18,19);(2,6,9,7)(4,12)(8,14,16,15)(10,13)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 463);