Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C33 |
| direct product | C5×C3⋊(C3⋊D3) |
| unambiguous semidirect product | C3⋊(C5×C3⋊D3) |
| Fitting, socle, Hall | C33×C5 ⋊ C2 |
| derived, Sylow | C33 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊(C3⋊D3) |
| commutator, cosocle, hypocenter | C33 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C5 | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 4 generators on 14 points:
(2,3)(5,6)(8,9)(10,11,12,13,14);(7,8,9);(4,5,6);(1,2,3)75 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C3⋊(C3⋊D3), C5×D3×C3⋊D3, D15⋊(C3⋊D3), … |
| maximal normal subgroup of | D5×C3⋊(C3⋊D3), C5×D3×C3⋊D3, D15⋊(C3⋊D3), … |
| maximal quotient of | C5×C3⋊(C3⋊Dic3), C2×C5×C3⋊(C3⋊D3), C5×C3⋊(C3⋊D9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 28);