Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5230 |
| Sylow | C3 ⋊ C2×#395 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C23×C3×Q8 | C2 |
| socle, prime radical | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | C2×#395 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
908 conjugacy classes of subgroups; only the 285 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 285 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:
(3,4,5)(6,7)(8,9)(14,17,18,21)(15,19,16,20);(1,2)(10,11)(12,13)(14,16,18,15)(17,19,21,20);(1,2)(4,5)(7,9)(11,13)(14,15,18,16)(17,19,21,20);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10690178, #10690183, #10695401, … |
| maximal normal subgroup of | #10690178, #10690183, #10695401, … |
| maximal quotient of | C2×#71566, C2×#71568, C2×#72134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16640);