Finite Group DatabasebetaFinite Group Databasebeta| order | 600 |
| isoclinism | isoclinic class of D5×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D5×A4 |
| unambiguous semidirect product | C5⋊(C2×C5×A4) |
| unambiguous semidirect product | (C22×D5)⋊(C3×C5) |
| unambiguous semidirect product | (C22×C5×D5)⋊C3 |
| Fitting, socle | C22×C52 ⋊ C2×C3 |
| derived | C22×C5 ⋊ C2×C3×C5 |
| Hall | C22×C5×D5 ⋊ C3 |
| Sylow | C52 ⋊ C2×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D5×A4 |
| commutator, cosocle, hypocenter | C22×C5 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C52 | C2×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C5 | |
58 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | C5×A4⋊AGL(1,𝔽5), A4⋊#1042, C5×A4×AGL(1,𝔽5), … |
| maximal normal subgroup of | C5×A4⋊AGL(1,𝔽5), A4⋊#1042, C5×A4×AGL(1,𝔽5), … |
| maximal quotient of | C5×C5⋊(#41⋊C3), C5×D5×SL(2,𝔽3), C5×A4×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(600, 181);