Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C2×C4×#157 |
| Fitting | C2×C32×C4 ⋊ C4 |
| derived, Sylow | C32 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #157 |
| commutator, hypocenter | C32 | C2×C42 |
| Frattini | C2 | C22×#157 |
| Fitting | C2×C32×C4 | C4 |
| socle | C22×C32 | C2×C4 |
| prime radical | C22×C3⋊D3 | C22 |
| cosocle | C2×C3⋊D3 | C23 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C4 | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(7,8)(9,10,11,12);(3,4,5,6)(7,11,9);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #10590044, #10590047, #10590090, … |
| maximal normal subgroup of | #10590044, #10590047, #10590090, … |
| maximal quotient of | C4×#3787, #10593723, C4×#3788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 932);