Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#559 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| prime radical | C2×C3×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5);(7,10)(11,12)(14,15,18,17)(16,20,21,19);(6,7,9,10)(8,12,13,11)(15,17)(19,20);(14,16,18,21)(15,19,17,20);(1,2)(6,8,9,13)(7,11,10,12)(14,17,18,15)(16,20,21,19)150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19939);