Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#2818⋊C3 |
| unambiguous semidirect product | (C2×#2818)⋊C3 |
| Fitting, Sylow | C2×#2818 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #560⋊C3 |
| commutator, cosocle | C2×C42 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2818 | C3 |
| socle | C24 | C22×A4 |
| hypocenter | C42 | C2×C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C22 | |
746 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 746 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213963, #72213984, C2×#46324⋊C3, … |
| maximal normal subgroup of | #72213963, #72213984, C2×#46324⋊C3, … |
| maximal quotient of | C2×#17091⋊C3, #72228380, C2×#14661⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084427);