Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊#16861523 |
| Fitting, socle, derived, Hall | C3×C52×C41 ⋊ C4 |
| Hall | C52×C41 ⋊ Dic3 |
| Hall | C3×C41 ⋊ #1044 |
| Hall | C3×C52 ⋊ Dic41 |
| Sylow | C41 ⋊ C3⋊#1044 |
| Sylow | C52 ⋊ Dic123 |
| Sylow | C3 ⋊ #16861523 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C52×C41 | C4 |
| prime radical, cosocle | C3×C41×C5⋊D5 | C2 |
| LCS | G ▷ C3×C52×C41 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,5^2,41|a^2=b,c^a=c^2,d^a=d^2,e^a=e^3,f^a=f^40,d^b=d^4,e^b=e^4864 conjugacy classes — view the full character table.
| maximal subgroup of | #49146806, C9⋊#16861523, C3⋊#27134245, … |
| maximal normal subgroup of | C3×C3⋊#16861523 |
| maximal quotient of | C9⋊#16861523, C3×C3⋊#16861523, C3⋊(C3⋊#16861523) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 165);