Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#73293 |
| Fitting, Sylow | C2×#2840 ⋊ C3 |
| derived | #2840 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | C22⋊A4 |
| commutator, cosocle, hypocenter | #2840 | C2×C3 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C2×#2840 | C3 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C24 | |
925 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 925 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72227587, C2×#11714257, #72227612, … |
| maximal normal subgroup of | C2×#11714257, #72227614, #72227693, … |
| maximal quotient of | #72227820, #72227874, #72227916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084417);