Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #392⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#1444⋊C3 |
| unambiguous semidirect product | (C2×#1444)⋊C3 |
| Fitting, Sylow | C2×#1444 ⋊ C3 |
| derived | #95 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #127⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini | C24 | C22×A4 |
| Fitting | C2×#1444 | C3 |
| socle | C25 | C2×A4 |
| prime radical | C22×#95⋊C3 | C2 |
| cosocle | C2×#95 | C22×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C23 | |
695 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#392⋊C3, #72214627, #72215419, … |
| maximal normal subgroup of | C2×C4×#392⋊C3, #72214627, #72215419, … |
| maximal quotient of | C2×#72211231, C2×#11337⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083608);