Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1437⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#1437⋊C3 |
| unambiguous semidirect product | (C2×#1437)⋊C3 |
| Fitting, Sylow | C2×#1437 ⋊ C3 |
| derived | C22×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1437⋊C3 |
| commutator, cosocle, hypocenter | C22×C42 | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | C2×#1437 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C2 | |
945 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 945 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #72222710, #72224446, #72225771, … |
| maximal normal subgroup of | C2×#15258⋊C3, C2×#11711026, C2×#11710955, … |
| maximal quotient of | C2×#10935⋊C3, C2×#7555⋊C3, C2×#7556⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083586);