Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1451 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C23 |
| commutator | C22 | C2×C3×C42 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C42 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ G | |
208 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,6,4,5,7,8)(10,13)(14,16)(17,18,19);(9,10,11,14,12,13,15,16);(2,5)(6,8)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4742);