Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of (C5⋊D5)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5 |
| direct product | C2×(C5⋊D5)⋊C3 |
| unambiguous semidirect product | (C2×C5⋊D5)⋊C3 |
| Fitting, socle | C2×C52 ⋊ C2×C3 |
| derived, Sylow | C52 ⋊ C22×C3 |
| Hall | C2×C5⋊D5 ⋊ C3 |
| Hall | C52⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C5⋊D5)⋊C3 |
| commutator, cosocle, hypocenter | C52 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C52 | C2×C3 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 17 points:
(3,4,6)(5,15,12,8,9,17)(7,11,16,13,14,10);(1,2)(4,7,9,15,13)(6,11,17,12,14)20 conjugacy classes — view the full character table.
| maximal subgroup of | #10597079, #10597080, C4×(C5⋊D5)⋊C3, … |
| maximal normal subgroup of | #10597079, #10597080, C4×(C5⋊D5)⋊C3, … |
| maximal quotient of | #6045⋊C3, C4×(C5⋊D5)⋊C3, C4⋊((C5⋊D5)⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 27);