Finite Group DatabasebetaFinite Group Databasebeta| order | 440 |
| isoclinism | isoclinic class of (C11⋊C5)⋊Q8 |
| automorphism | Aut(G) = D4×AGL(1,𝔽11) (order 880) · Inn(G) = C2×AGL(1,𝔽11) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C11⋊C5)⋊Q8 |
| unambiguous semidirect product | Dic22⋊C5 |
| Hall | Dic22 ⋊ C5 |
| Hall | C11⋊C5 ⋊ Q8 |
| Sylow | C11 ⋊ C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽11) |
| commutator, socle, cosocle | C2×C11 | C22×C5 |
| Fitting | C4×C11 | C2×C5 |
| prime radical | C2×C11⋊C5 | C22 |
| hypercenter | C4 | AGL(1,𝔽11) |
| hypocenter | C11 | C5×Q8 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 88 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,3,4,8)(2,5,6,7)(9,10,11,16,19)(12,17,13,18,15);(1,2,4,6)(3,7,8,5)(10,11)(12,13)(14,15)(16,18)(17,19)Polycyclic presentation:
2^2,5,2,11|a^2=d,b^2=d,b^a=bd,e^a=e^10,e^c=e^929 conjugacy classes — view the full character table.
| maximal subgroup of | #4314⋊C5, Dic44⋊C5, #4324⋊C5, … |
| maximal normal subgroup of | #4314⋊C5, Dic44⋊C5, #4324⋊C5, … |
| maximal quotient of | (Dic11⋊C4)⋊C5, C4⋊(Dic11⋊C5), C2×(C11⋊C5)⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(440, 7);