Finite Group DatabasebetaFinite Group Databasebeta| order | 8724 |
| automorphism | Aut(G) order 6333624 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D4362 |
| direct product | C2×D2181 |
| unambiguous semidirect product | (C2×C3)⋊D727 |
| unambiguous semidirect product | (C2×C727)⋊D3 |
| Fitting, socle | C2×C3×C727 ⋊ C2 |
| derived, Hall | C3×C727 ⋊ C22 |
| Sylow | C727 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D727 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D2181 |
| commutator, hypocenter | C3×C727 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C727 | C2 |
| LCS | G ▷ C3×C727 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
2184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D6543, C2×C3⋊(D727⋊C3), D3×D2181, … |
| maximal normal subgroup of | C2×C3⋊(D727⋊C3), C2×C3×D2181, D8724 |
| maximal quotient of | C2×D6543, D3×D2181, C2×C3×D2181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8724, 13);