Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#381 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1144 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 1144 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) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5);(7,9)(10,11,12,13)(15,18)(19,21);(1,2)(14,15)(16,19)(17,18)(20,21);(6,7)(8,9)(10,11)(12,13)(14,16,17,20)(15,19,18,21)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16719);