Finite Group DatabasebetaFinite Group Databasebeta| order | 1170 |
| isoclinism | isoclinic class of D13⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×D13⋊C9 |
| unambiguous semidirect product | D13⋊(C5×C9) |
| unambiguous semidirect product | (C13⋊C9)⋊(C2×C5) |
| unambiguous semidirect product | (C5×D13)⋊C9 |
| derived, Sylow | C13 ⋊ C2×C5×C9 |
| Hall | C5×C13⋊C9 ⋊ C2 |
| Hall | C5×D13 ⋊ C9 |
| Hall | C13⋊C9 ⋊ C2×C5 |
| Hall | C5×C13 ⋊ C2×C9 |
| Hall | D13 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | D13⋊C3 |
| commutator, hypocenter | C13 | C2×C5×C9 |
| Frattini | C3 | C5×D13⋊C3 |
| Fitting, socle | C3×C5×C13 | C2×C3 |
| prime radical | C3×C5×D13 | C3 |
| cosocle | C3×C13 | C2×C3×C5 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C3×C5 | |
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)(6,10,14,8,13,9,11,7,12)(16,20,21,17,18,22)(19,26,25,23,24,27);(15,16,19,18,22,25,24,26,27,21,20,23,17)120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#261⋊C9, #63118⋊C9, D5×D13⋊C9, … |
| maximal normal subgroup of | C5×#261⋊C9, #63118⋊C9, D5×D13⋊C9, … |
| maximal quotient of | C5×Dic13⋊C9, C2×C5×D13⋊C9, C5×C11×D13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1170, 1);