Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D32×D4 |
| Fitting | C32×D4 ⋊ C22 |
| Sylow | C32 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×D4 | C22 |
| hypercenter | D4 | D32 |
| hypocenter | C32 | C22×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
499 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 10 points:
(2,4)(5,6,7)(8,9,10);(1,2)(3,4);(9,10);(6,7)45 conjugacy classes — view the full character table.
| maximal subgroup of | #10593457, #10593459, D32×D8, … |
| maximal normal subgroup of | #10593457, #10593459, D32×D8, … |
| maximal quotient of | D3×#878, #10594054, #10594055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 958);