Finite Group DatabasebetaFinite Group Databasebeta| order | 440 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×C5×AGL(1,𝔽5) (order 400) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22×C5 |
| homology | Abelianization (H1) = C8×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#170 |
| derived, Sylow | C5 ⋊ C8×C11 |
| Hall | C5×C11 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C8×C11 |
| Frattini | C2 | C11×AGL(1,𝔽5) |
| Fitting, socle, prime radical | C2×C5×C11 | C4 |
| cosocle | Dic5 | C2×C11 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C11 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 440 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,4,3)(6,10,12,7,9,13,8,11)(14,15,16,17,18,19,20,21,22,23,24);(1,2,3,5,4)110 conjugacy classes — view the full character table.
| maximal subgroup of | D11×#170, #11724960, #11724961, … |
| maximal normal subgroup of | D11×#170, #11724960, #11724961, … |
| maximal quotient of | C11×#674, C2×C11×#170, C3×C11×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(440, 15);