Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | A4×#638 |
| Fitting | C24×C32 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×A4×C3⋊D3 |
| commutator, socle, cosocle | C23×C32 | C22×C3 |
| Fitting | C24×C32 | C2×C3 |
| hypercenter | C22 | A4×C3⋊D3 |
| hypocenter | C22×C32 | C3×D4 |
| LCS | G ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
476 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #13180781, #13180782, S4×#638, … |
| maximal normal subgroup of | #13180781, #13180782, S4×#638, … |
| maximal quotient of | A4×#3750, A4×#3752, A4×#3753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4462);