Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×#87551 |
| Fitting, Sylow | C2×2−1+6 ⋊ C3 |
| derived | 2−1+6 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | C22⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | 2−1+6 | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊A4) |
| Fitting | C2×2−1+6 | C3 |
| LCS | G ▷ 2−1+6 | |
| UCS | C1 ◁ C22 | |
10061 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 10061 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11710348, C2×#72210027, C2×#11710938, … |
| maximal normal subgroup of | C2×#11710348, C2×#11710938, C22×#87551, … |
| maximal quotient of | C2×#72210038, C2×#11710852, C22×#87551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085316);