Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#3026⋊C3 |
| unambiguous semidirect product | (C2×#3026)⋊C3 |
| Fitting, Sylow | C2×#3026 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#3026 | C3 |
| hypercenter | C23 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22 ◁ C23 | |
1038 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 1038 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 |
46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72209831, C2×#11714170, #72227791, … |
| maximal normal subgroup of | C2×#11714170, #72227797, C2×#11714182, … |
| maximal quotient of | D3×#3026⋊C3, #72245412, C4×#3026⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084723);