Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of A4×#1042 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | A4×#1042 |
| unambiguous semidirect product | (C22×#1042)⋊C3 |
| Fitting, socle, derived | C22×C52 ⋊ C3×C4 |
| Hall | C22×#1042 ⋊ C3 |
| Sylow | C52 ⋊ C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C52 | C3×C4 |
| prime radical | C5×D5×A4 | C2 |
| cosocle | C22×C5×D5 | C2×C3 |
| LCS | G ▷ C22×C52 | |
| UCS | C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 80 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | #72222081, D5×A4×AGL(1,𝔽5), S4×#1042, … |
| maximal normal subgroup of | D5×A4×AGL(1,𝔽5), S4×#1042, #72258513 |
| maximal quotient of | #72222081 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 978);