Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#1791⋊C3 |
| unambiguous semidirect product | (C2×#1791)⋊C3 |
| Fitting, Sylow | C2×#1791 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C4 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | C2×#1791 | C3 |
| socle | C22 | #379⋊C3 |
| prime radical | C2×#5950 | C2 |
| cosocle | #585 | C22×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C4 | |
682 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 682 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20612⋊C3, C2×#11711596, C2×#11711597, … |
| maximal normal subgroup of | C2×#20612⋊C3, C2×#11711596, C2×#11711597, … |
| maximal quotient of | #72342113 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083842);