Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×C8×Dic3 |
| derived, Sylow | C3 ⋊ C22×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C8 | D3 |
| commutator, hypocenter | C3 | C22×C4×C8 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C23×C3×C8 | C2 |
| socle, prime radical | C24×C3 | C2×C4 |
| cosocle | C2×C3×C4 | C24 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C8 | |
760 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 549 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(3,4)(6,7)(8,9,10,11);(5,6,7)(12,13,14,16,15,17,18,19);(3,4);(1,2)192 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#336, #10700338, #10700342, … |
| maximal normal subgroup of | Dic3×#336, #10700338, #10700342, … |
| maximal quotient of | C22×C8×C3⋊C8, C22×#4454, C2×C8×#208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16300);