Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×#5934 (order 7680) · Inn(G) = D5 · Out(G) = C22×#5934 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×Dic5 |
| derived, Sylow | C5 ⋊ C22×C3×C4 |
| Hall | C3×C5 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D5 |
| commutator, hypocenter | C5 | C22×C3×C4 |
| Frattini | C2 | C22×C3×D5 |
| Fitting, socle, prime radical | C23×C3×C5 | C2 |
| cosocle | C2×C5 | C23×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C3 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(5,6,7)(8,9,10,11)(13,14)(15,16);(3,4)(12,13,15,16,14);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #90994, C3×#3963, C3×C22⋊#170, … |
| maximal normal subgroup of | #90994, C3×#3963, C3×C22⋊#170, … |
| maximal quotient of | C22×C3×#168, C2×C3×#681, C2×C3×C4×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 163);