Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#3025⋊C3 |
| unambiguous semidirect product | (C2×#3025)⋊C3 |
| Fitting, Sylow | C2×#3025 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#3025 | C3 |
| socle | C22 | C2×#120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C4 | |
986 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 986 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | #72227429, C2×#11714175, C2×#11714162, … |
| maximal normal subgroup of | C2×#11714175, C2×#11714162, #72236792, … |
| maximal quotient of | #72245692, C2×#23467⋊C3, D3×#3025⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084722);