Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#2275 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C2 | C22×C3×C42 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
780 conjugacy classes of subgroups; only the 520 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 520 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 — 4 generators on 19 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,19);(1,2,3,4)(9,11)(10,13)(12,16)(14,15);(5,7,8,6)(11,16)(13,15);(10,14)(13,15)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16715);