Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C3×C4×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C22 | C25×C3 |
| socle | C22×C3 | C25 |
| prime radical | C2×C3×2−1+4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
1492 conjugacy classes of subgroups; only the 1372 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1372 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3,4)(21,22,23);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(6,12)(10,13)(11,20)(18,19)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19890);