Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = C22×A4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×D4×A4 |
| unambiguous semidirect product | C4⋊(C2×C3×A4) |
| unambiguous semidirect product | (C22×D4)⋊C32 |
| unambiguous semidirect product | (C22×C3×D4)⋊C3 |
| Fitting | C22×C3×D4 ⋊ C3 |
| Sylow | C22×D4 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×A4 |
| commutator, cosocle | C23 | C22×C32 |
| Frattini | C2 | C22×C3×A4 |
| Fitting | C22×C3×D4 | C3 |
| socle | C23×C3 | C22×C3 |
| hypercenter | C3×D4 | A4 |
| hypocenter | C22 | C32×D4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 11 points:
(2,3,4)(6,8);(1,4)(2,3)(5,6)(7,8)(9,10,11)60 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#5414, C3×#5415, #10593648, … |
| maximal normal subgroup of | C3×#5414, C3×#5415, #10593648, … |
| maximal quotient of | C3×A4×C22⋊C4, C3×A4×C4⋊C4, C3×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 980);