Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of C5⋊S4 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = C5⋊S4 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×C5⋊S4 |
| unambiguous semidirect product | (C2×C5)⋊S4 |
| unambiguous semidirect product | (C2×A4)⋊D5 |
| Fitting, socle | C23×C5 ⋊ D3 |
| derived | C5×A4 ⋊ C22 |
| Sylow | C5 ⋊ C2×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊S4 |
| commutator, cosocle, hypocenter | C5×A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | D3 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(3,4,6,5)(8,9)(10,11);(1,2)(4,5,6)(7,8,10,11,9)26 conjugacy classes — view the full character table.
| maximal subgroup of | #91901, #91902, #91905, … |
| maximal normal subgroup of | #91901, #91902, #91905, … |
| maximal quotient of | #91948, C4×C5⋊S4, C4⋊(C5⋊S4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 197);