Finite Group DatabasebetaFinite Group Databasebeta| order | 12404 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×D443 |
| unambiguous semidirect product | C443⋊(C22×C7) |
| unambiguous semidirect product | (C2×C443)⋊(C2×C7) |
| Fitting, socle | C2×C7×C443 ⋊ C2 |
| derived, Sylow | C443 ⋊ C22×C7 |
| Hall | C7×C443 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D443 |
| commutator, cosocle, hypocenter | C443 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C443 | C2 |
| LCS | G ▷ C443 | |
| UCS | C1 ◁ C2×C7 | |
20 conjugacy classes of subgroups.
3122 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×ΓL(1,𝔽8)×D443, D3×C7×D443, C2×C3×C7×D443, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×D443, C2×C3×C7×D443 |
| maximal quotient of | D3×C7×D443, C2×C3×C7×D443, C2×C7×D1329 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12404, 9);