Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of C5⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×C5⋊S4 |
| unambiguous semidirect product | (C2×C5)⋊(C5×S4) |
| unambiguous semidirect product | (C2×A4)⋊(C5×D5) |
| Fitting, socle | C23×C52 ⋊ D3 |
| derived | C5×A4 ⋊ C22×C5 |
| Sylow | C52 ⋊ C2×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C5⋊S4 |
| commutator, cosocle, hypocenter | C5×A4 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | D3 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C2×C5 | |
156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
130 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C5×C5⋊S4, C2×(C5⋊S4)⋊D5, C2×C5×D5×S4, … |
| maximal normal subgroup of | C22×C5×C5⋊S4, C2×(C5⋊S4)⋊D5, C2×C5×D5×S4, … |
| maximal quotient of | C22×C5×C5⋊S4, C5×#91958 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 1009);