Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C32×2−1+4⋊C3 |
| unambiguous semidirect product | 2−1+4⋊C33 |
| unambiguous semidirect product | (C3×2−1+4)⋊C32 |
| Fitting | C32×2−1+4 ⋊ C3 |
| Sylow | 2−1+4 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C33 |
| Frattini | C2 | C22×C32×A4 |
| Fitting | C32×2−1+4 | C3 |
| hypercenter | C32×D4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C32 ◁ C32×D4 | |
484 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
171 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#5430, C32×#5431, C3×#10593664, … |
| maximal normal subgroup of | C32×#5430, C32×#5431, C3×#10593664, … |
| maximal quotient of | C32×#519⋊C3, C32×#520⋊C3, C32×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4607);