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 | C2×C4×D3×C8 |
| Fitting | C2×C3×C4×C8 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C8 | D3 |
| commutator, hypocenter | C3 | C22×C4×C8 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C2×C3×C4×C8 | C2 |
| socle | C23×C3 | C22×C4 |
| prime radical | C23×D3 | C2×C4 |
| cosocle | C2×C3×C4 | C24 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C4×C8 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(4,5);(3,4,5)(10,11,12,14,13,15,16,17);(6,7,8,9);(1,2)192 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1273, #10684567, #10684571, … |
| maximal normal subgroup of | D3×#1273, #10684567, #10684571, … |
| maximal quotient of | C2×D3×C82, C2×D3×#322, D3×C8×M16, … |
| automorphism group of | C5×C3⋊C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11025);