Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #4625⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#4625⋊C2 |
| unambiguous semidirect product | #4625⋊C22 |
| unambiguous semidirect product | (#1020⋊C2)⋊C22 |
| unambiguous semidirect product | #5450⋊C22 |
| Fitting | C2×H(3,𝔽4) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625⋊C2 |
| commutator, cosocle | #120⋊C3 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C2×H(3,𝔽4) | D3 |
| socle | C23 | GL(2,ℤ/4) |
| hypocenter | C22⋊A4 | C2×D4 |
| LCS | G ▷ #120⋊C3 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
564 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088544);