Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | D3×#3719 |
| Sylow | C33 ⋊ C2×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#626 |
| commutator, Fitting, cosocle | C33×C4 | C23 |
| Frattini, hypercenter | C4 | D33 |
| socle | C2×C33 | C2×D4 |
| prime radical | C2×C32×D3 | D4 |
| hypocenter | C33 | C2×Q16 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
326 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | D32×Dic12, D3×#63800, D3×#63801, … |
| maximal normal subgroup of | D32×Dic12, D3×#63800, D3×#63801, … |
| maximal quotient of | D3×#63566, D3×#63567, D3×#63575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2812);