Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#383 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(6,7,8,9)(11,14)(12,16)(19,22)(23,25);(10,11)(12,15)(13,14)(16,17)(18,19,21,22)(20,23,24,25);(1,2)(10,12,13,16)(11,15,14,17)(18,20,21,24)(19,23,22,25)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16722);