Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×#365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #106 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C23×C3 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
152 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(7,11,12,17)(8,14)(9,16)(13,20,19,18)(15,21);(6,7,8,11,9,13,14,18,10,12,15,17,16,19,21,20)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5445);