Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C22×C33 |
| direct product | C32×D4×A4 |
| unambiguous semidirect product | C4⋊(C2×C32×A4) |
| unambiguous semidirect product | (C22×D4)⋊C33 |
| unambiguous semidirect product | (C22×C3×D4)⋊C32 |
| Fitting | C22×C32×D4 ⋊ C3 |
| Sylow | C22×D4 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C22×A4 |
| commutator, cosocle | C23 | C22×C33 |
| Frattini | C2 | C22×C32×A4 |
| Fitting | C22×C32×D4 | C3 |
| socle | C23×C32 | C22×C3 |
| hypercenter | C32×D4 | A4 |
| hypocenter | C22 | C33×D4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2×C32 ◁ C32×D4 | |
652 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#5414, C32×#5415, C3×#10593648, … |
| maximal normal subgroup of | C32×#5414, C32×#5415, C3×#10593648, … |
| maximal quotient of | C32×A4×C22⋊C4, C32×A4×C4⋊C4, C32×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4602);