Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = D3×A4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×Dic3×A4 |
| unambiguous semidirect product | (C2×C3)⋊(C4×A4) |
| unambiguous semidirect product | (C23×Dic3)⋊C3 |
| derived | C22×C3 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×A4 |
| commutator, hypocenter | C22×C3 | C2×C3×C4 |
| Frattini | C2 | C2×D3×A4 |
| Fitting, socle | C24×C3 | C2×C3 |
| prime radical | C22×C3×A4 | C2 |
| cosocle | C23×C3 | C22×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C22 | |
142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(4,5)(7,8,9)(10,11,12,13);(1,2)(3,4,5)(6,9)(7,8)48 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×SL(2,ℤ/4), #10593238, SL(2,ℤ/4)⋊Dic3, … |
| maximal normal subgroup of | Dic3×SL(2,ℤ/4), #10593238, SL(2,ℤ/4)⋊Dic3, … |
| maximal quotient of | C2×A4×C3⋊C8, A4×#208, C4×Dic3×A4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 927);