Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#85566 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020 |
| commutator, hypocenter | #997 | C2×C4 |
| Frattini | C4 | C2×#1020 |
| Fitting | C2×#585 | D3 |
| socle | C22 | #5936 |
| prime radical | C2×#5950 | C2 |
| cosocle | #5950 | C22 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
911 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 911 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11715687, C2×#72268571, #72268582, … |
| maximal normal subgroup of | C2×#11715687, #72268582, C2×#11715661, … |
| maximal quotient of | #72342124 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090041);