Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#72994 |
| Fitting | C2×#384 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857⋊C2 |
| commutator, cosocle | C2×C42⋊C3 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C2×#384 | D3 |
| socle | C24 | C2×S4 |
| hypocenter | C42⋊C3 | C2×D4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 | |
734 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 40 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 | C2×#72213198, #72215431, #72218186, … |
| maximal normal subgroup of | #72218186, #72218204, #72226168, … |
| maximal quotient of | C2×#11711335, #72213885, #72226154 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087529);