Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#2205⋊C3 |
| unambiguous semidirect product | (C2×#2205)⋊C3 |
| Fitting, Sylow | C2×#2205 ⋊ C3 |
| derived | Q82 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | Q82 | C22×C3 |
| Frattini | C2×Q8 | C22×A4 |
| Fitting | C2×#2205 | C3 |
| hypercenter | C23 | #120⋊C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C22 ◁ C23 | |
286 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | #72227401, #72227699, #72228023, … |
| maximal normal subgroup of | #72228026, #72268125, C22×#2205⋊C3 |
| maximal quotient of | #72227546, C2×#72212285, C2×#72210824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084089);