Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #2029⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#2029⋊C3 |
| unambiguous semidirect product | (C2×#2029)⋊C3 |
| Fitting, Sylow | C2×#2029 ⋊ C3 |
| derived | #543 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #120⋊C3 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | C2×#2029 | C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C23 | |
718 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14515304, #72209935, #72227293, … |
| maximal normal subgroup of | #72209935, C2×#11711413, #72227715, … |
| maximal quotient of | #72227305, #72227739, #72227888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083906);