Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C22 |
| commutator | C2 | C23×C3×C8 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
440 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 392 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) = 384 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4,5,6,7,8,9,10);(11,12,13,14)(15,18,17,16);(11,15,13,17)(12,16,14,18)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17398);