Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×C4×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | C22 |
| commutator | C2 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C23×C3 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ G | |
196 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 176 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 — 3 generators on 23 points:
(1,2,3)(4,5,6,7);(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);(9,17)(11,19)(13,21)(15,23)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5398);