Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of A4⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×A4⋊AGL(1,𝔽5) |
| unambiguous semidirect product | A4⋊(C5×AGL(1,𝔽5)) |
| Fitting, socle | C22×C52 ⋊ Dic3 |
| derived | C5×A4 ⋊ C4×C5 |
| Sylow | C52 ⋊ SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | A4⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×A4 | C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C52 | Dic3 |
| prime radical | C5×D5×A4 | C2 |
| cosocle | D5×A4 | C2×C5 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C5 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 80 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#11757174, D5×A4⋊AGL(1,𝔽5), #72228881, … |
| maximal normal subgroup of | D5×A4⋊AGL(1,𝔽5), C5×AGL(1,𝔽5)×S4, #72258513 |
| maximal quotient of | C5×#11757174, #72228881 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 969);