Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C42 |
| direct product | C42×C42⋊C3 |
| Fitting, Sylow | C44 ⋊ C3 |
| derived | C42 ⋊ C3×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | C42⋊C3 |
| commutator, hypocenter | C42 | C3×C42 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | C44 | C3 |
| prime radical | C22×C42⋊C3 | C22 |
| cosocle | C22×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C42 | |
736 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | #72209724, #72209727, #72210939, … |
| maximal normal subgroup of | #72209724, C2×C42×C42⋊C3, C4×#1842⋊C3, … |
| maximal quotient of | C2×C42×C42⋊C3, #72222564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083579);