Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C4×A4×C3⋊D3 |
| Fitting | C22×C32×C4 ⋊ C2×C3 |
| derived | C22×C32 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4×C3⋊D3 |
| commutator, hypocenter | C22×C32 | C2×C3×C4 |
| Frattini | C2 | C2×A4×C3⋊D3 |
| Fitting | C22×C32×C4 | C2×C3 |
| socle, cosocle | C23×C32 | C22×C3 |
| prime radical | C2×A4×C3⋊D3 | C2 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C4 | |
444 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×A4⋊C8, #13165932, #13166076, … |
| maximal normal subgroup of | C3⋊D3×A4⋊C8, #13165932, #13166076, … |
| maximal quotient of | C4×C3⋊(C3⋊(#41⋊C3)), #13173143, C4×C3⋊D3×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4453);