Finite Group DatabasebetaFinite Group Databasebeta| order | 12220 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C47⋊#63122 |
| Fitting, socle, derived, Hall | C5×C13×C47 ⋊ C4 |
| Hall | C13×C47 ⋊ AGL(1,𝔽5) |
| Hall | C5×C47 ⋊ #261 |
| Hall | C5×C13 ⋊ Dic47 |
| Sylow | C47 ⋊ #63122 |
| Sylow | C13 ⋊ C47⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ C47⋊#261 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C13×C47 | C4 |
| prime radical, cosocle | C47×D65 | C2 |
| LCS | G ▷ C5×C13×C47 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,5,13,47|a^2=b,c^a=c^2,d^a=d^5,e^a=e^46,c^b=c^4,d^b=d^12802 conjugacy classes — view the full character table.
| maximal subgroup of | C47⋊(#63122⋊C3), C3×C47⋊#63122, C3⋊(C47⋊#63122) |
| maximal normal subgroup of | C47⋊(#63122⋊C3), C3×C47⋊#63122 |
| maximal quotient of | C3×C47⋊#63122, C3⋊(C47⋊#63122) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12220, 16);