Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C22×#458⋊C3 |
| Fitting, Sylow | C22×#458 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C22×#458 | C3 |
| prime radical | C22×#997 | C2 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 | |
1034 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 64 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 | C22×#3026⋊C3, C22×#73030, C22×#73032, … |
| maximal normal subgroup of | C22×#3026⋊C3, C22×#73030, C22×#73032, … |
| maximal quotient of | C2×#32976⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084720);