Finite Group DatabasebetaFinite Group Databasebeta| order | 600 |
| isoclinism | isoclinic class of #6045 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5 |
| direct product | C3×C4⋊(C5⋊D5) |
| unambiguous semidirect product | C4⋊(C3×C5⋊D5) |
| Fitting | C3×C4×C52 ⋊ C2 |
| Hall | C3×C52 ⋊ D4 |
| Sylow | C52 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×C5⋊D5 |
| commutator, cosocle | C2×C52 | C22×C3 |
| Frattini | C2 | C2×C3×C5⋊D5 |
| Fitting | C3×C4×C52 | C2 |
| socle | C2×C3×C52 | C22 |
| hypercenter | C3×C4 | C5⋊D5 |
| hypocenter | C52 | C3×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 300 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3)(4,5,6,7)(13,14,16,17,15);(5,7)(9,10)(11,12)(14,15)(16,17);(8,9,11,12,10)159 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#87771, C3×#87774, #11925105, … |
| maximal normal subgroup of | C3×#87771, C3×#87774, #11925105, … |
| maximal quotient of | C3×#87800, C3×#87801, C3×#87802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(600, 123);