Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D32×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×#624 |
| unambiguous semidirect product | (C22×#624)⋊C3 |
| derived | C22×C32 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×A4 |
| commutator, hypocenter | C22×C32 | C2×C3×C4 |
| Fitting, socle, cosocle | C23×C32 | C22×C3 |
| prime radical | C2×A4×C3⋊D3 | C2 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
335 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #13180213, #13180214, A4×D32⋊C4, … |
| maximal normal subgroup of | #13180213, #13180214, A4×D32⋊C4, … |
| maximal quotient of | A4×#3710, A4×#3712, #10595386⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4191);