Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×#120 (order 3840) · Inn(G) = D15 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | C4×Dic15 |
| derived, Hall | C3×C5 ⋊ C42 |
| Sylow | C5 ⋊ C4×Dic3 |
| Sylow | C3 ⋊ C4×Dic5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D15 |
| commutator, hypocenter | C3×C5 | C42 |
| Frattini | C22 | C2×D15 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle, prime radical, cosocle | C22×C3×C5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5)(7,8)(13,14,15,16);(1,2,4,5,3)(6,7,8)(9,10,11,12)72 conjugacy classes — view the full character table.
| maximal subgroup of | #90951, #90954, #90976, … |
| maximal normal subgroup of | #90951, #90954, #90976, … |
| maximal quotient of | C4×C3⋊#168, #91087, C8×Dic15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 72);