Finite Group DatabasebetaFinite Group Databasebeta| order | 6762 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | C3⋊(C7⋊D161) |
| unambiguous semidirect product | C7⋊D483 |
| unambiguous semidirect product | C23⋊(C3⋊(C7⋊D7)) |
| Fitting, socle, derived, Hall | C3×C72×C23 ⋊ C2 |
| Hall | C72×C23 ⋊ D3 |
| Hall | C3×C72 ⋊ D23 |
| Hall | C3×C23 ⋊ C7⋊D7 |
| Sylow | C72 ⋊ D69 |
| Sylow | C23 ⋊ C3⋊(C7⋊D7) |
| Sylow | C3 ⋊ C7⋊D161 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C72×C23 | C2 |
| LCS | G ▷ C3×C72×C23 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7^2,23|b^a=b^2,c^a=c^6,d^a=d^6,e^a=e^221692 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D23×C3⋊(C7⋊D7), C7⋊D7×D69, D3×C7⋊D161, … |
| maximal normal subgroup of | D23×C3⋊(C7⋊D7), C7⋊D7×D69, D3×C7⋊D161, … |
| maximal quotient of | C3⋊(C7⋊Dic161), C2×C3⋊(C7⋊D161), C7⋊D1449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6762, 46);