Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic30 |
| automorphism | Aut(G) = C2×D3×D4×AGL(1,𝔽5) (order 1920) · Inn(G) = C2×D15 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×C4×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ #41 |
| Sylow | C5 ⋊ #235 |
| Sylow | C3 ⋊ #709 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D15 |
| commutator, socle, cosocle | C2×C3×C5 | C23 |
| Frattini | C2 | C22×D15 |
| Fitting | C2×C3×C4×C5 | C2 |
| hypercenter | C2×C4 | D15 |
| hypocenter | C3×C5 | #41 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C4 ◁ C2×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)(6,7,8)(10,14)(13,16);(9,11,12,15)(10,13,14,16);(2,3)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^4,3,5|b^2=d,c^a=cd,e^a=e^2,f^a=f^466 conjugacy classes — view the full character table.
| maximal subgroup of | #90980, #90981, #91089, … |
| maximal normal subgroup of | #90980, #90981, #91089, … |
| maximal quotient of | C4×Dic30, #91759, #91761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 178);