Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C22×C8 |
| direct product | D3×C8×C16 |
| Fitting | C3×C8×C16 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C8×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8×C16 | D3 |
| commutator, hypocenter | C3 | C2×C8×C16 |
| Frattini | C4×C8 | C22×D3 |
| Fitting | C3×C8×C16 | C2 |
| socle | C22×C3 | C2×C4×C8 |
| prime radical | C22×D3 | C4×C8 |
| cosocle | C3×C4×C8 | C23 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C8×C16 | |
398 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
384 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×C8×M32, D3×#12517, D3×#12515, … |
| maximal normal subgroup of | D3×C8×M32, D3×#12517, D3×#12515, … |
| maximal quotient of | D3×#7052, D3×C162 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 78538);