Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C2×C4×D3×AGL(1,𝔽5) (order 960) · Inn(G) = D15 · Out(G) = C2×C42 |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic15 |
| unambiguous semidirect product | C3⋊(C5×Dic5) |
| derived | C3×C5 ⋊ C4×C5 |
| Hall | C3×C52 ⋊ C4 |
| Sylow | C52 ⋊ Dic3 |
| Sylow | C3 ⋊ C5×Dic5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D15 |
| commutator, hypocenter | C3×C5 | C4×C5 |
| Frattini | C2 | C5×D15 |
| Fitting, socle, prime radical | C2×C3×C52 | C2 |
| cosocle | C2×C3×C5 | C2×C5 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C5 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,4,5)(7,8)(9,10)(12,13)(14,15,16,17);(6,7,9,10,8)(11,12,13)90 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D5×Dic3, C5×D3×Dic5, C5×#1201, … |
| maximal normal subgroup of | C5×D5×Dic3, C5×D3×Dic5, C5×#1201, … |
| maximal quotient of | C5×C3⋊#168, C2×C5×Dic15, C5×Dic45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 19);