Finite Group DatabasebetaFinite Group Databasebeta| order | 1170 |
| isoclinism | isoclinic class of D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×D45 |
| unambiguous semidirect product | C5⋊(C13×D9) |
| unambiguous semidirect product | C9⋊(D5×C13) |
| Fitting, Hall | C5×C9×C13 ⋊ C2 |
| derived, Hall | C5×C9 ⋊ C2×C13 |
| Hall | C9×C13 ⋊ D5 |
| Hall | C5×C13 ⋊ D9 |
| Sylow | C9 ⋊ D5×C13 |
| Sylow | C5 ⋊ C13×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C13 | D45 |
| commutator, cosocle, hypocenter | C5×C9 | C2×C13 |
| Frattini | C3 | C13×D15 |
| Fitting | C5×C9×C13 | C2 |
| socle | C3×C5×C13 | D3 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C13 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 585 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22)(23,24,25,26,27);(15,22)(16,21)(17,20)(18,19)(24,27)(25,26)312 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D13×D45, D5×C13×D9, D45⋊D13, … |
| maximal normal subgroup of | D13×D45, D5×C13×D9, D45⋊D13, … |
| maximal quotient of | C13×Dic45, C2×C13×D45, C11×C13×D45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1170, 10);