Finite Group DatabasebetaFinite Group Databasebeta| order | 10004 |
| automorphism | Aut(G) order 12004800 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D5002 |
| direct product | C2×D2501 |
| unambiguous semidirect product | (C2×C41)⋊D61 |
| unambiguous semidirect product | (C2×C61)⋊D41 |
| Fitting, socle | C2×C41×C61 ⋊ C2 |
| derived, Hall | C41×C61 ⋊ C22 |
| Sylow | C61 ⋊ C2×D41 |
| Sylow | C41 ⋊ C2×D61 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D2501 |
| commutator, cosocle, hypocenter | C41×C61 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C41×C61 | C2 |
| LCS | G ▷ C41×C61 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
2504 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C41⋊(D61⋊C3), D3×D2501, C2×C3×D2501, … |
| maximal normal subgroup of | C2×C41⋊(D61⋊C3), C2×C3×D2501, D10004 |
| maximal quotient of | D3×D2501, C2×C3×D2501, C2×D7503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10004, 13);