Finite Group DatabasebetaFinite Group Databasebeta| order | 6996 |
| automorphism | Aut(G) order 3637920 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D3498 |
| direct product | C2×D1749 |
| unambiguous semidirect product | (C2×C3)⋊D583 |
| unambiguous semidirect product | (C2×C11)⋊D159 |
| unambiguous semidirect product | (C2×C53)⋊D33 |
| Fitting, socle | C2×C3×C11×C53 ⋊ C2 |
| derived, Hall | C3×C11×C53 ⋊ C22 |
| Hall | C11×C53 ⋊ C2×D3 |
| Hall | C3×C53 ⋊ C2×D11 |
| Sylow | C53 ⋊ C2×D33 |
| Hall | C3×C11 ⋊ C2×D53 |
| Sylow | C11 ⋊ C2×D159 |
| Sylow | C3 ⋊ C2×D583 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1749 |
| commutator, hypocenter | C3×C11×C53 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C11×C53 | C2 |
| LCS | G ▷ C3×C11×C53 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
1752 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D5247, D3×D1749, C2×C3×D1749, … |
| maximal normal subgroup of | C2×C3×D1749, C2×C3⋊(C53⋊AGL(1,𝔽11)), C2×C5×D1749, … |
| maximal quotient of | C2×D5247, D3×D1749, C2×C3×D1749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6996, 22);