Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of C3⋊(C5⋊D5) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C5 |
| direct product | C2×C3⋊(C5⋊D5) |
| unambiguous semidirect product | (C2×C3)⋊(C5⋊D5) |
| unambiguous semidirect product | (C2×C5)⋊D15 |
| Fitting, socle | C2×C3×C52 ⋊ C2 |
| derived, Hall | C3×C52 ⋊ C22 |
| Sylow | C52 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C5⋊D5) |
| commutator, cosocle, hypocenter | C3×C52 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C52 | C2 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C2 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 150 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(4,5)(7,8)(9,10)(12,13)(14,15);(1,2)(3,4,5)(6,7,9,10,8);(11,12,14,15,13)78 conjugacy classes — view the full character table.
| maximal subgroup of | #10597113, #10597115, #10597116, … |
| maximal normal subgroup of | #10597113, #10597115, #10597116, … |
| maximal quotient of | #10597155, C4×C3⋊(C5⋊D5), C4⋊(C3⋊(C5⋊D5)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 48);