Finite Group DatabasebetaFinite Group Databasebeta| order | 600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | C4⋊((C5⋊D5)⋊C3) |
| unambiguous semidirect product | (C4⋊(C5⋊D5))⋊C3 |
| Fitting | C4×C52 ⋊ C2×C3 |
| Hall | C4⋊(C5⋊D5) ⋊ C3 |
| Hall | C52⋊C3 ⋊ D4 |
| Sylow | C52 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×(C5⋊D5)⋊C3 |
| commutator, socle, cosocle | C2×C52 | C22×C3 |
| Fitting | C4×C52 | C2×C3 |
| hypercenter | C4 | (C5⋊D5)⋊C3 |
| hypocenter | C52 | C3×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 60 |
Polycyclic presentation:
2^2,3,2,5^2|b^2=d,b^a=bd,e^a=e^4,f^a=f^4,e^c=e^3f^3,f^c=e^4f31 conjugacy classes — view the full character table.
| maximal subgroup of | #11925009, #11925010, #87800⋊C3, … |
| maximal normal subgroup of | #11925009, #11925010, #87800⋊C3, … |
| maximal quotient of | #87800⋊C3, #87801⋊C3, #87802⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(600, 69);