Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#2165 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #117 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | #432 |
| prime radical | C23×C3 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#97 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,8,4,9,11,16,5,6,10,13,12,7,14,15)(17,18,19);(2,6)(3,10)(7,9)(11,14);(2,7,6,9)(3,10)(8,16,13,15)(11,14)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5456);