Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C22 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C4 | C25×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
1504 conjugacy classes of subgroups; only the 1384 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 1384 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)(5,6,7);(3,4);(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23);(10,16)(13,19)(17,22)(20,23);(1,2)(3,4)(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20031);