Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#73286 |
| Fitting, Sylow | C2×#2276 ⋊ C3 |
| derived | #2276 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C22⋊(C42⋊C3) |
| commutator, cosocle, hypocenter | #2276 | C2×C3 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C2×#2276 | C3 |
| socle | C24 | C22⋊A4 |
| LCS | G ▷ #2276 | |
| UCS | C1 ◁ C22 | |
674 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 674 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11714240, C2×#11714241, #72239183, … |
| maximal normal subgroup of | C2×#11714240, C2×#11714241, #72239183 |
| maximal quotient of | C2×#72222569, C2×#13834⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084134);