Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #396⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×#396⋊C3 |
| unambiguous semidirect product | (C4×#396)⋊C3 |
| Fitting, Sylow | C4×#396 ⋊ C3 |
| derived | #95 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #126⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | C4×#396 | C3 |
| prime radical | C2×#396⋊C3 | C2 |
| cosocle | C2×#95 | C22×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2×C4 | |
431 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #72222701, #72222703, #72222820, … |
| maximal normal subgroup of | #72222701, #72222703, #72222820, … |
| maximal quotient of | #72222696, #72222699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083617);