Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of C5⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C5 |
| 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 | C52×A4 ⋊ C22 |
| Sylow | C52 ⋊ C2×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C52×A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | D3 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
264 conjugacy classes of subgroups.
| transitive degree | μt(G) = 150 |
106 conjugacy classes — view the full character table.
| maximal subgroup of | #72230064, C2×S4×C5⋊D5, #72230250, … |
| maximal normal subgroup of | #72230064, C2×S4×C5⋊D5, #72230250, … |
| maximal quotient of | #72230064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 1010);