Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×#103 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C23×C3×C4 ◁ G | |
816 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 440 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) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(5,6,7);(3,4);(8,9,10,11)(13,16)(14,18)(17,19);(8,9)(10,11)(12,13,15,17)(14,19,18,16)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17331);