Finite Group DatabasebetaFinite Group Databasebeta| order | 27342 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×C3⋊((C31⋊C3)⋊D7) |
| unambiguous semidirect product | C3⋊(C7×(C31⋊C3)⋊D7) |
| unambiguous semidirect product | (C31⋊C3)⋊(C7×D21) |
| Fitting, socle | C3×C72×C31 ⋊ C2×C3 |
| derived | C3×C7×C31 ⋊ C2×C3×C7 |
| Hall | C3×C72×C31⋊C3 ⋊ C2 |
| Hall | C72×C31 ⋊ C3×D3 |
| Hall | C3×C31⋊C3 ⋊ C7×D7 |
| Sylow | C72 ⋊ C3⋊(D31⋊C3) |
| Sylow | C31 ⋊ C3×C7×D21 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C3⋊((C31⋊C3)⋊D7) |
| commutator, cosocle, hypocenter | C3×C7×C31 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72×C31 | C2×C3 |
| LCS | G ▷ C3×C7×C31 | |
| UCS | C1 ◁ C7 | |
108 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1302 |
987 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27342, 303);