Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C2×D5 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#709 |
| Fitting | C2×C3×C4×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ #41 |
| Sylow | C5 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D5 |
| commutator, cosocle | C2×C5 | C23×C3 |
| Frattini | C2 | C22×C3×D5 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle | C2×C3×C5 | C23 |
| hypercenter | C2×C3×C4 | D5 |
| hypocenter | C5 | C3×#41 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,5,3)(7,11)(10,13)(14,15,16);(6,12,9,8)(7,13,11,10);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)78 conjugacy classes — view the full character table.
| maximal subgroup of | #90976, #90977, C3×#3937, … |
| maximal normal subgroup of | #90976, #90977, C3×#3937, … |
| maximal quotient of | C3×C4×Dic10, C3×#4016, C3×#4018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 158);