Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#72988 |
| Fitting | C2×#376 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C2×#376 | D3 |
| socle | C24 | C2×S4 |
| hypercenter | C2×C4 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
642 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #72218276, #72227453, C2×#11711943, … |
| maximal normal subgroup of | #72218276, #72227453, C2×#11711943, … |
| maximal quotient of | C2×#11711005, C2×#72213081, #72227448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086198);