Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | D3×#3724 |
| Sylow | C33 ⋊ C2×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D33 |
| commutator | C2×C33 | C22×C4 |
| Fitting, socle, cosocle | C22×C33 | C23 |
| prime radical | C22×C32×D3 | C22 |
| hypocenter | C33 | C2×C4⋊C4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
490 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63739, D3×#63740, #13165467, … |
| maximal normal subgroup of | D3×#63739, D3×#63740, #13165467, … |
| maximal quotient of | D3×#63572, #13155447, #13155455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2876);