Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×#1020 (order 11520) · Inn(G) = D3×D5 · Out(G) = C2×#1020 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×D5 |
| unambiguous semidirect product | (C5×D3)⋊C23 |
| unambiguous semidirect product | (C3×D5)⋊C23 |
| unambiguous semidirect product | D15⋊C23 |
| Fitting, socle | C22×C3×C5 ⋊ C22 |
| derived, Hall | C3×C5 ⋊ C24 |
| Sylow | C5 ⋊ C23×D3 |
| Sylow | C3 ⋊ C23×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D5 |
| commutator, cosocle, hypocenter | C3×C5 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4)(5,6,7)(9,10)(11,12);(6,7)(8,9,11,12,10);(3,4);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C2×D3)⋊C4, D3×#685, #91473, … |
| maximal normal subgroup of | D5×(C2×D3)⋊C4, D3×#685, #91473, … |
| maximal quotient of | C2×D5×Dic6, C2×#6765, C2×#6766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 202);