Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C22×D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D3×#237 |
| Fitting | C32×D4 ⋊ C22 |
| Sylow | C32 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×D4 | C22 |
| hypercenter | D4 | D32 |
| hypocenter | C32 | C2×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
348 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(2,6)(3,7)(9,10,11)(12,13,14);(1,3,4,7)(2,5,6,8);(1,2)(3,5)(4,6)(7,8)(13,14);(10,11)45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#911, D3×#912, #10594728, … |
| maximal normal subgroup of | D3×#911, D3×#912, #10594728, … |
| maximal quotient of | #10594025, #10594026, #10594027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 959);