Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C24 |
| direct product | C3×#1304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #95 |
| commutator | C23 | C3×C42 |
| Frattini, cosocle | C25 | C22×C3 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C25×C3 ◁ G | |
396 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)(3,7,8,5)(4,6)(9,10)(11,15)(12,14)(13,16)(17,18,19);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13,14,15)(12,16)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 440);