Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #338⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C4×#338⋊C3 |
| unambiguous semidirect product | (C4×#338)⋊C3 |
| Fitting, Sylow | C4×#338 ⋊ C3 |
| derived | #338 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | C42⋊C3 |
| commutator, hypocenter | #338 | C3×C4 |
| Frattini | C23×C4 | C2×A4 |
| Fitting | C4×#338 | C3 |
| socle | C22 | C2×#95⋊C3 |
| prime radical | C2×#338⋊C3 | C2 |
| cosocle | C2×#338 | C2×C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C42 | |
198 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 29 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 | #72209692, C2×C4×#338⋊C3, C4×#2017⋊C3, … |
| maximal normal subgroup of | C2×C4×#338⋊C3, C4×#2017⋊C3, #72213108 |
| maximal quotient of | C2×C4×#338⋊C3, #72210939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083543);