Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | D3×SL(2,ℤ/4) |
| unambiguous semidirect product | (D3×A4)⋊C4 |
| derived | C3×A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S4 |
| commutator, hypocenter | C3×A4 | C2×C4 |
| Fitting, socle | C23×C3 | C2×D3 |
| prime radical | C2×D3×A4 | C2 |
| cosocle | C2×C3×A4 | C22 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
132 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,3,4)(5,6,8,7)(9,10,11);(6,7,8)(10,11)30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×A4⋊Q8, #10596424, #10596425, … |
| maximal normal subgroup of | D3×A4⋊Q8, #10596424, #10596425, … |
| maximal quotient of | D3×A4⋊C8, #10593182, #10593191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 856);