Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| isoclinism | isoclinic class of C5⋊((C3⋊D3)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C5⋊((C3⋊D3)⋊C3) |
| unambiguous semidirect product | (C3⋊D15)⋊C3 |
| Fitting, Hall | C5×H(3,𝔽3) ⋊ C2 |
| derived | C32×C5 ⋊ C2×C3 |
| Sylow | H(3,𝔽3) ⋊ D5 |
| Sylow | C5 ⋊ (C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C5 | C2×C3 |
| Frattini | C3 | C3×D15 |
| Fitting | C5×H(3,𝔽3) | C2 |
| socle | C3×C5 | C3×D3 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,5)(7,12,14,9,10,13)(8,11);(1,2,4,5,3)(6,7,9)(8,10,13)(11,14,12)Polycyclic presentation:
2,3^3,5|c^a=c^2,d^a=d^2,e^a=e^4,c^b=cd32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 14);