Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#2032⋊C3 |
| unambiguous semidirect product | (C2×#2032)⋊C3 |
| Fitting, Sylow | C2×#2032 ⋊ C3 |
| derived | #543 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393⋊C3 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | C2×#2032 | C3 |
| hypercenter | C23 | #120⋊C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22 ◁ C23 | |
354 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72209831, #72227732, #72227791, … |
| maximal normal subgroup of | C4×#2032⋊C3, C2×#23474⋊C3, C2×#32873⋊C3, … |
| maximal quotient of | D3×#2032⋊C3, C4×#2032⋊C3, D5×#2032⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083912);