Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#2825⋊C3 |
| unambiguous semidirect product | (C2×#2825)⋊C3 |
| Fitting, Sylow | C2×#2825 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #560⋊C3 |
| commutator, cosocle, hypocenter | #95 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2825 | C3 |
| socle | C24 | C22×A4 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 | |
605 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 605 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72222820, #72222829, #72222832, … |
| maximal normal subgroup of | #72222820, #72222829, #72222832 |
| maximal quotient of | C2×#17091⋊C3, C2×#72260709 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084414);