Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C4 | C25×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5);(6,7,9,13,10,12,16,19)(8,11,15,18,14,17,20,21);(8,14)(11,17)(15,20)(18,21);(6,8)(7,11)(9,15)(10,14)(12,17)(13,18)(16,20)(19,21);(1,2)(7,12)(11,17)(13,19)(18,21)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20032);