Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×A4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×D3×A4 |
| unambiguous semidirect product | (C22×C4×D3)⋊C3 |
| Fitting | C22×C3×C4 ⋊ C2×C3 |
| derived | C22×C3 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×A4 |
| commutator, hypocenter | C22×C3 | C2×C3×C4 |
| Frattini | C2 | C2×D3×A4 |
| Fitting | C22×C3×C4 | C2×C3 |
| socle, cosocle | C23×C3 | C22×C3 |
| prime radical | C2×D3×A4 | C2 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C4 | |
138 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 33 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:
(6,7,8)(10,11);(1,2,3,4)(5,8)(6,7)(9,10,11)48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×A4⋊C8, #10593182, D3×C8×A4, … |
| maximal normal subgroup of | D3×A4⋊C8, #10593182, D3×C8×A4, … |
| maximal quotient of | C4×C3⋊(#41⋊C3), #5574⋊C3, C4×D3×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 919);