Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×#41 (order 384) · Inn(G) = D3 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C3⋊C16 |
| unambiguous semidirect product | (C2×C3)⋊(C3×C16) |
| derived | C3 ⋊ C2×C3×C16 |
| Sylow | C32 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C8 | D3 |
| commutator, hypocenter | C3 | C2×C3×C16 |
| Frattini | C8 | C2×C3×D3 |
| Fitting | C2×C32×C8 | C2 |
| socle, prime radical | C22×C32 | C8 |
| cosocle | C3×C8 | C22×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C8 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(4,5)(6,7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2)(3,4,5)144 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C3⋊C16, #10589102, #10589103, … |
| maximal normal subgroup of | Dic3×C3⋊C16, #10589102, #10589103, … |
| maximal quotient of | C3×C4×C3⋊C16, C3×C4⋊(C3⋊C16), C2×C3×C3⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 245);