Finite Group DatabasebetaFinite Group Databasebeta| order | 600 |
| isoclinism | isoclinic class of C5⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×C5⋊S4 |
| unambiguous semidirect product | C5⋊(C5×S4) |
| unambiguous semidirect product | A4⋊(C5×D5) |
| Fitting, socle | C22×C52 ⋊ D3 |
| derived | C5×A4 ⋊ C2×C5 |
| Sylow | C52 ⋊ S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C5⋊S4 |
| commutator, cosocle, hypocenter | C5×A4 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C52 | D3 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C5 | |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
65 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D5×S4, D5×C5⋊S4, (C5⋊S4)⋊D5, … |
| maximal normal subgroup of | C5×D5×S4, D5×C5⋊S4, (C5⋊S4)⋊D5, … |
| maximal quotient of | C5×C5⋊#226, C5×C5⋊GL(2,𝔽3), C5×A4⋊Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(600, 177);