Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #327 |
| commutator | C2×C8 | C2×C3×C4 |
| Frattini, cosocle | C2×M16 | C22×C3 |
| prime radical | C3×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
110 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,9,6,10,5,14)(3,7,11,8,13,16,12,15)(17,18,19);(2,7)(3,11)(4,5)(8,9)(10,16)(12,13)(14,15)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 477);