Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic30 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×#120 (order 3840) · Inn(G) = C2×D15 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊Dic15 |
| Hall | C3×C5 ⋊ C4⋊C4 |
| Sylow | C5 ⋊ C4⋊Dic3 |
| Sylow | C3 ⋊ C4⋊Dic5 |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,5,3)(6,7,8)(13,14,15,16);(2,3)(4,5)(7,8)(9,10,11,12)(14,16)Polycyclic presentation:
2^4,3,5|a^2=c,b^2=d,b^a=bd,e^a=e^2,f^a=f^466 conjugacy classes — view the full character table.
| maximal subgroup of | #90965, #90967, #90971, … |
| maximal normal subgroup of | #90965, #90967, #90971, … |
| maximal quotient of | C4⋊(C3⋊#168), #91101, #91102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 74);