Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of C82⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C8 |
| direct product | C4×C82⋊C3 |
| unambiguous semidirect product | (C4×C82)⋊C3 |
| Fitting, Sylow | C4×C82 ⋊ C3 |
| derived | C82 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C82⋊C3 |
| commutator, hypocenter | C82 | C3×C4 |
| Frattini | C2×C42 | C2×A4 |
| Fitting | C4×C82 | C3 |
| socle | C23 | C2×C42⋊C3 |
| prime radical | C2×C82⋊C3 | C2 |
| cosocle | C2×C82 | C2×C3 |
| LCS | G ▷ C82 | |
| UCS | C1 ◁ C4 | |
145 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | #72210254, C2×C4×C82⋊C3, #72210757, … |
| maximal normal subgroup of | C2×C4×C82⋊C3, #72210757, #72210759 |
| maximal quotient of | #72210252, C2×C4×C82⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083489);