Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic30 |
| automorphism | Aut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = C2×D15 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C5×C8 ⋊ C2 |
| Hall | C3×C5 ⋊ M16 |
| Sylow | C5 ⋊ ΓL(1,𝔽25) |
| Sylow | C3 ⋊ #676 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D15 |
| commutator, socle | C2×C3×C5 | C2×C4 |
| Fitting | C3×C5×C8 | C2 |
| prime radical | C2×D15 | C4 |
| cosocle | C3×C4×C5 | C22 |
| hypercenter | C8 | D15 |
| hypocenter | C3×C5 | M16 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C4 ◁ C8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,5,3)(6,7,8)(9,10,11,14,12,13,15,16);(2,3)(4,5)(7,8)(10,13)(14,16)Polycyclic presentation:
2^4,3,5|b^2=c,c^2=d,b^a=bd,e^a=e^2,f^a=f^466 conjugacy classes — view the full character table.
| maximal subgroup of | D5×ΓL(1,𝔽25), D3×#676, #91257, … |
| maximal normal subgroup of | D5×ΓL(1,𝔽25), D3×#676, #91257, … |
| maximal quotient of | #91098, #91099, #91104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 66);