Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C4×D3×GL(2,𝔽3) (order 1152) · Inn(G) = D3 · Out(G) = C4×GL(2,𝔽3) |
| homology | Abelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3 |
| direct product | C32×C5×D3 |
| unambiguous semidirect product | C3⋊(C2×C32×C5) |
| Fitting, socle, Hall | C33×C5 ⋊ C2 |
| derived | C3 ⋊ C2×C32×C5 |
| Sylow | C33 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C5 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C32×C5 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 14 points:
(4,5,6)(8,9)(10,11,12,13,14);(1,2,3)(7,8,9)135 conjugacy classes — view the full character table.
| maximal subgroup of | C32×D3×D5, C3×C5×D32, C3×D3×D15, … |
| maximal normal subgroup of | C32×D3×D5, C3×C5×D32, C3×D3×D15, … |
| maximal quotient of | C32×C5×Dic3, C2×C32×C5×D3, C32×C5×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 24);