Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #16861520⋊C3 |
| Fitting, socle, derived, Hall | C52×C41 ⋊ C3×C4 |
| Hall | #16861520 ⋊ C3 |
| Hall | C41×C52⋊C3 ⋊ C4 |
| Hall | C52⋊C3 ⋊ Dic41 |
| Sylow | C41 ⋊ #1043⋊C3 |
| Sylow | C52 ⋊ C3×Dic41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C41 | C3×C4 |
| prime radical | C41×(C5⋊D5)⋊C3 | C2 |
| cosocle | C41×C5⋊D5 | C2×C3 |
| LCS | G ▷ C52×C41 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,2,5^2,41|a^2=c,d^a=d^3,e^a=e^3,f^a=f^40,d^b=d^3e^3,e^b=d^4e,d^c=d^4,e^c=e^4214 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#16861520⋊C3, C3⋊(#16861520⋊C3) |
| maximal normal subgroup of | C3×#16861520⋊C3 |
| maximal quotient of | #16861520⋊C9, C3×#16861520⋊C3, C3⋊(#16861520⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 104);