Finite Group DatabasebetaFinite Group Databasebeta| order | 440 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C2×C5×D4×AGL(1,𝔽5) (order 1600) · Inn(G) = C2×D5 · Out(G) = C22×C4×C5 |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×D20 |
| unambiguous semidirect product | C4⋊(D5×C11) |
| Fitting | C4×C5×C11 ⋊ C2 |
| Hall | C5×C11 ⋊ D4 |
| Sylow | C5 ⋊ D4×C11 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C11 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C11 |
| Frattini | C2 | C2×D5×C11 |
| Fitting | C4×C5×C11 | C2 |
| socle | C2×C5×C11 | C22 |
| hypercenter | C4×C11 | D5 |
| hypocenter | C5 | D4×C11 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C11 ◁ C4×C11 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 220 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,4,3,2)(5,6,8,9,7)(10,11,12,13,14,15,16,17,18,19,20);(2,4)(6,7)(8,9)143 conjugacy classes — view the full character table.
| maximal subgroup of | #11724871, #11724873, #11724875, … |
| maximal normal subgroup of | #11724871, #11724873, #11724875, … |
| maximal quotient of | C11×#677, C11×D40, C11×Dic20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(440, 31);