Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | #107 |
| prime radical | C23×C3 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(3,4,5);(6,7,10,15,12,16,11,26)(8,17,20,32,22,33,21,36)(9,23,24,30,25,37,18,14)(13,27,28,35,29,34,31,19);(6,8,18,34,11,21,25,35,12,22,24,27,10,20,9,19)(7,13,23,32,15,28,30,33,16,29,37,36,26,31,14,17)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5451);