Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1273⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C4 |
| direct product | C2×#1273⋊C3 |
| unambiguous semidirect product | (C2×#1273)⋊C3 |
| Fitting, Sylow | C2×#1273 ⋊ C3 |
| derived | #1273 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C82⋊C3 |
| commutator, cosocle, hypocenter | #1273 | C2×C3 |
| Frattini | C2×C42 | C2×A4 |
| Fitting | C2×#1273 | C3 |
| socle | C24 | C42⋊C3 |
| LCS | G ▷ #1273 | |
| UCS | C1 ◁ C22 | |
192 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | #72210223, C2×#8131⋊C3, C22×#1273⋊C3, … |
| maximal normal subgroup of | #72210223, C2×#8131⋊C3, C22×#1273⋊C3, … |
| maximal quotient of | C2×#7014⋊C3, C2×#72210253, C22×#1273⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083492);