Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#346 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C22 |
| commutator | C2 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(3,4,5);(10,11,12,16,13,17,18,23,14,15,19,21,20,22,24,25);(6,7,8,9)(11,15)(16,21)(17,22)(23,25)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5397);