Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| automorphism | Aut(G) = C2×C4×D3×D4 (order 384) · Inn(G) = C22×D3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×D3×D4 |
| Fitting | C3×C5×D4 ⋊ C2 |
| Hall | C3×C5 ⋊ C2×D4 |
| Sylow | C3 ⋊ C2×C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C22×D3 |
| commutator, cosocle | C2×C3 | C23×C5 |
| Frattini | C2 | C22×C5×D3 |
| Fitting | C3×C5×D4 | C2 |
| socle | C2×C3×C5 | C23 |
| hypercenter | C5×D4 | D3 |
| hypocenter | C3 | C2×C5×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C5 ◁ C5×D4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,7)(8,9,10,11,12);(1,2)(3,4);(6,7)75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#686, D3×#687, #91494, … |
| maximal normal subgroup of | D3×#686, D3×#687, #91494, … |
| maximal quotient of | C5×#878, C5×D3×C22⋊C4, C5×#67⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 169);