Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽5) (order 320) · Inn(G) = D5 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×Dic5 |
| derived | C5 ⋊ C3×C4×C5 |
| Hall | C3×C52 ⋊ C4 |
| Sylow | C52 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | D5 |
| commutator, hypocenter | C5 | C3×C4×C5 |
| Frattini | C2 | C3×C5×D5 |
| Fitting, socle, prime radical | C2×C3×C52 | C2 |
| cosocle | C2×C5 | C2×C3×C5 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×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)(11,14,13,12)(15,16,17);(6,7,9,10,8)120 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×#170, C3×#6032, C5×C3⋊#170, … |
| maximal normal subgroup of | C3×C5×#170, C3×#6032, C5×C3⋊#170, … |
| maximal quotient of | C3×C5×#168, C2×C3×C5×Dic5, C5×C9×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 16);