Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C4×D3×#157 |
| Fitting | C33×C4 ⋊ C2×C4 |
| derived, Sylow | C33 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×#157 |
| commutator, hypocenter | C33 | C2×C42 |
| Frattini | C2 | C2×D3×#157 |
| Fitting | C33×C4 | C2×C4 |
| socle | C2×C33 | C22×C4 |
| prime radical | C2×D3×C3⋊D3 | C22 |
| cosocle | C2×C3×C3⋊D3 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C4 | |
356 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63731, #13165163, D3×#63732, … |
| maximal normal subgroup of | D3×#63731, #13165163, D3×#63732, … |
| maximal quotient of | D3×C8×#157, #157×ΓL(1,𝔽25), D3×#63767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4223);