Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×#87543 |
| Fitting, Sylow | C23×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C23×C3 |
| Frattini | C2 | C23×C22⋊A4 |
| Fitting | C23×H(5,𝔽2) | C3 |
| socle | C24 | C22×A4 |
| hypercenter | C2×Q8 | C22⋊A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
11151 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 11151 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72231801, #72231954 |
| maximal normal subgroup of | #72231954 |
| maximal quotient of | C2×#72231801 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085303);