Finite Group DatabasebetaFinite Group Databasebeta| order | 550 |
| isoclinism | isoclinic class of C5⋊D55 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C5⋊D55 |
| unambiguous semidirect product | C11⋊(C5⋊D5) |
| Fitting, socle, derived, Hall | C52×C11 ⋊ C2 |
| Sylow | C52 ⋊ D11 |
| Sylow | C11 ⋊ C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C11 | C2 |
| LCS | G ▷ C52×C11 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 275 |
Polycyclic presentation:
2,5^2,11|b^a=b^4,c^a=c^4,d^a=d^10139 conjugacy classes — view the full character table.
| maximal subgroup of | D11×C5⋊D5, D5×D55, C2×C5⋊D55, … |
| maximal normal subgroup of | D11×C5⋊D5, D5×D55, C2×C5⋊D55, … |
| maximal quotient of | C11⋊#1039, C2×C5⋊D55, C3×C5⋊D55, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(550, 15);