Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C3×D4×S4 (order 576) · Inn(G) = C22×A4 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
| direct product | D4×C22⋊C9 |
| unambiguous semidirect product | C4⋊(C2×C22⋊C9) |
| unambiguous semidirect product | (C22×D4)⋊C9 |
| Sylow | C22×D4 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C3 | C22×A4 |
| commutator | C23 | C22×C9 |
| Fitting, prime radical | C22×C3×D4 | C3 |
| socle, cosocle | C23×C3 | C22×C3 |
| hypercenter | C3×D4 | A4 |
| hypocenter | C22 | D4×C9 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,5,9,3,8,4,6,2,7)(11,13)(15,16,17);(10,11)(12,13)(14,17)(15,16)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10589516, #10589517, D8×C22⋊C9, … |
| maximal normal subgroup of | #10589516, #10589517, D8×C22⋊C9, … |
| maximal quotient of | C22⋊C4×C22⋊C9, C4⋊C4×C22⋊C9, #548⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 344);