Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | A4×#627 |
| unambiguous semidirect product | (C22×#627)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×A4 |
| commutator, Fitting, socle, cosocle | C23×C32 | C22×C3 |
| prime radical | C2×C32×A4 | C22 |
| hypocenter | C22×C32 | C3×Q8 |
| LCS | G ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
235 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #13180217, #13180218, A4×#3775, … |
| maximal normal subgroup of | #13180217, #13180218, A4×#3775, … |
| maximal quotient of | #10595422⋊C3, #10595351⋊C3, SL(2,𝔽3)×#627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4194);