Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#405 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C22 |
| commutator | C2 | C22×C3×C42 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(10,11,13,16,14,17,20,23)(12,15,19,22,18,21,24,25);(6,7,8,9)(12,18)(15,21)(19,24)(22,25);(1,2)(10,12)(11,15)(13,19)(14,18)(16,22)(17,21)(20,24)(23,25)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17312);