Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊#16861528 |
| Fitting, socle, derived, Hall | C3×C52×C41 ⋊ C4 |
| Hall | C52×C41 ⋊ Dic3 |
| Hall | C3×C41 ⋊ #1039 |
| Hall | C3×C52 ⋊ #4223 |
| Sylow | C41 ⋊ C3⋊#1039 |
| Sylow | C52 ⋊ C3⋊#4223 |
| Sylow | C3 ⋊ #16861528 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C52×C41 | C4 |
| prime radical, cosocle | C3×C52×D41 | C2 |
| LCS | G ▷ C3×C52×C41 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,5^2,41|a^2=b,c^a=c^2,d^a=d^4,e^a=e^4,f^a=f^32,f^b=f^40828 conjugacy classes — view the full character table.
| maximal subgroup of | C9⋊#16861528, C3⋊(#16861528⋊C3), C3×C3⋊#16861528, … |
| maximal normal subgroup of | C3⋊(#16861528⋊C3), C3×C3⋊#16861528 |
| maximal quotient of | C9⋊#16861528, C3×C3⋊#16861528, C3⋊(C3⋊#16861528) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 170);