Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of C22⋊(C22⋊A4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×C22⋊(C22⋊A4) |
| Fitting, Sylow | C26×C4 ⋊ C3 |
| derived | C26 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C22⋊(C22⋊A4) |
| commutator, hypocenter | C26 | C3×C4 |
| Frattini | C2 | C2×C22⋊(C22⋊A4) |
| Fitting | C26×C4 | C3 |
| socle, cosocle | C27 | C2×C3 |
| prime radical | C2×C22⋊(C22⋊A4) | C2 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C4 | |
18753 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×C22⋊(C22⋊A4), C4×#85645, C4×#87588, … |
| maximal normal subgroup of | C4×#85645, C4×#87588, C4×#10596782, … |
| maximal quotient of | C4×C22⋊#997, AGL(1,𝔽5)×C22⋊(C22⋊A4), #72342885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085286);