Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×C4×D3×D4 (order 384) · Inn(G) = C2×D3 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#235 |
| Fitting | C2×C3×C4×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ #41 |
| Sylow | C3 ⋊ C5×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C5 |
| Frattini | C2 | C22×C5×D3 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle | C2×C3×C5 | C23 |
| hypercenter | C2×C4×C5 | D3 |
| hypocenter | C3 | C5×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 | |
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:
(1,2,3)(5,9)(8,11)(12,13,14,15,16);(4,6,7,10)(5,8,9,11);(2,3)(4,5)(6,8)(7,9)(10,11)90 conjugacy classes — view the full character table.
| maximal subgroup of | #90978, #90979, C5×#805, … |
| maximal normal subgroup of | #90978, #90979, C5×#805, … |
| maximal quotient of | C4×C5×Dic6, C5×#870, C5×#872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 168);