Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#87521 |
| Fitting | C2×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C23 |
| Frattini | C2 | C22×#1020 |
| Fitting | C2×#585 | D3 |
| hypercenter | C2×C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1707 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 1707 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11715839, #72227944, D3×#87521, … |
| maximal normal subgroup of | C2×#11715839, #72227944, C2×#11714175, … |
| maximal quotient of | D3×#87521, #72240901, #72241291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090217);