Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C2×C52×D3 |
| unambiguous semidirect product | C3⋊(C22×C52) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C52) |
| Fitting, socle | C2×C3×C52 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C52 |
| Hall | C3×C52 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C52 | D3 |
| commutator, cosocle, hypocenter | C3 | C22×C52 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C52 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C52 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 150 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(4,5)(6,7,8,9,10);(1,2)(3,4,5)(11,12,13,14,15)150 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×Dic5, C5×#1201, C5×#1203, … |
| maximal normal subgroup of | C5×D3×Dic5, C5×#1201, C5×#1203, … |
| maximal quotient of | C52×Dic6, C4×C52×D3, C52×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 46);