Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #338⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C22×#338⋊C3 |
| Fitting, Sylow | C22×#338 ⋊ C3 |
| derived | #338 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C42⋊C3 |
| commutator, cosocle, hypocenter | #338 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | C22×#338 | C3 |
| socle | C23 | #95⋊C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C22×C4 | |
450 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #72209697, C2×C4×#338⋊C3, C22×#2017⋊C3, … |
| maximal normal subgroup of | C2×C4×#338⋊C3, C22×#2017⋊C3, C2×#11337⋊C3, … |
| maximal quotient of | C2×C4×#338⋊C3, C22×#7014⋊C3, C2×#72210936 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083804);