Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C22×C3×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(5,6,7);(3,4);(8,9)(10,14)(11,12)(13,15)(16,17)(18,20)(19,21)(22,23);(9,12)(13,14)(16,18,19,22)(17,20,21,23);(1,2)(3,4)(8,10,11,15)(9,13,12,14)(17,21)(20,23)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19898);