Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C4×#127⋊C3 |
| unambiguous semidirect product | (C2×C4×#127)⋊C3 |
| Fitting, Sylow | C2×C4×#127 ⋊ C3 |
| derived | C42 ⋊ C22×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #127⋊C3 |
| commutator, hypocenter | C42 | C22×C3×C4 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×C4×#127 | C3 |
| socle | C24 | C22×A4 |
| prime radical | C22×#127⋊C3 | C2 |
| cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2×C4 | |
1084 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72211680, C2×#72211684, #72218181, … |
| maximal normal subgroup of | #72218181, C4×#73084, C42×#127⋊C3, … |
| maximal quotient of | C2×C4×#384⋊C3, C2×#72211680, C2×C4×#392⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084162);