Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×C4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42 | C22 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C22 | C25×C3 |
| socle | C23×C3 | C24 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42 ◁ G | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 17 points:
(1,2)(3,4,5);(6,7,8,9);(11,14)(15,17);(10,11)(12,15)(13,14)(16,17);(1,2)(10,12,13,16)(11,15,14,17)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19884);