Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #3444⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×#3444⋊C3 |
| unambiguous semidirect product | (C2×#3444)⋊C3 |
| Fitting, Sylow | C2×#3444 ⋊ C3 |
| derived | C22×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3444⋊C3 |
| commutator, cosocle, hypocenter | C22×C42 | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C2×#3444 | C3 |
| socle | C23 | C2×C22⋊A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C2 | |
1493 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1493 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11714269, C2×#72209985, C2×#62832⋊C3, … |
| maximal normal subgroup of | C2×#11714269, C2×#62832⋊C3, C2×#11715691, … |
| maximal quotient of | C2×#72213869, C2×#35924⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085006);