Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×D3×C8 |
| Fitting | C2×C32×C8 ⋊ C2 |
| derived | C3 ⋊ C22×C3×C8 |
| Sylow | C32 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C8 | D3 |
| commutator, hypocenter | C3 | C22×C3×C8 |
| Frattini | C4 | C22×C3×D3 |
| Fitting | C2×C32×C8 | C2 |
| socle | C22×C32 | C2×C4 |
| prime radical | C22×C3×D3 | C4 |
| cosocle | C3×C4 | C23×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C8 | |
163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(4,5)(6,7,8);(3,4,5)(9,10,11,13,12,14,15,16);(1,2)144 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D3)⋊(C3⋊C16), C3×(C2×D3)⋊C16, C2×D3×C3⋊C16, … |
| maximal normal subgroup of | (C2×D3)⋊(C3⋊C16), C3×(C2×D3)⋊C16, C2×D3×C3⋊C16, … |
| maximal quotient of | C3×C8×Dic6, C3×C4×D3×C8, C3×#4685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 670);