Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#16861480 |
| Fitting, derived, Hall | C3×C25×C41 ⋊ C4 |
| Hall | C25×C41 ⋊ Dic3 |
| Hall | C3×C41 ⋊ #1035 |
| Hall | C3×C25 ⋊ #4223 |
| Sylow | C41 ⋊ C3⋊#1035 |
| Sylow | C25 ⋊ C3⋊#4223 |
| Sylow | C3 ⋊ #16861480 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C25×C41 | C4 |
| Frattini | C5 | C3⋊#11717961 |
| socle | C3×C5×C41 | AGL(1,𝔽5) |
| prime radical, cosocle | C3×D1025 | C2 |
| LCS | G ▷ C3×C25×C41 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,5,41,5|a^2=b,d^5=f,c^a=c^2,d^a=d^2f,e^a=e^32,f^a=f^2,d^b=d^4f^4,e^b=e^40,f^b=f^4774 conjugacy classes — view the full character table.
| maximal subgroup of | C9⋊#16861480, C3×C3⋊#16861480, C3⋊(C3⋊#16861480) |
| maximal normal subgroup of | C3×C3⋊#16861480 |
| maximal quotient of | C9⋊#16861480, C3×C3⋊#16861480, C3⋊(C3⋊#16861480) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 32);