Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1304⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#1304⋊C3 |
| unambiguous semidirect product | (C2×#1304)⋊C3 |
| Fitting, Sylow | C2×#1304 ⋊ C3 |
| derived | #1304 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1304⋊C3 |
| commutator, cosocle, hypocenter | #1304 | C2×C3 |
| Frattini | C25 | C2×A4 |
| Fitting | C2×#1304 | C3 |
| socle | C23 | #95⋊C3 |
| LCS | G ▷ #1304 | |
| UCS | C1 ◁ C2 | |
464 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
28 conjugacy classes — view the full character table.
| maximal subgroup of | #72210077, #72210079, #72210091, … |
| maximal normal subgroup of | C2×#8101⋊C3, C2×#11710964, C2×#11710960, … |
| maximal quotient of | C2×#7317⋊C3, #72227397, #72235536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083495);