Finite Group DatabasebetaFinite Group Databasebeta| order | 27468 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)⋊#89137 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3×#89137) |
| Fitting, socle | C3×C7×C109 ⋊ C3×C4 |
| derived, Hall | C7×C109 ⋊ C32×C4 |
| Hall | C3×ΓL(1,𝔽8)×C109 ⋊ C4 |
| Hall | C7⋊#89137 ⋊ C32 |
| Sylow | C109 ⋊ C3×ΓL(1,𝔽8)⋊C4 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #89137 |
| Sylow | C7 ⋊ C32×#89137 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)⋊#89137 |
| commutator, hypocenter | C7×C109 | C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C109 | C3×C4 |
| prime radical | C3×ΓL(1,𝔽8)×D109 | C2 |
| cosocle | C7×D109 | C2×C32 |
| LCS | G ▷ C7×C109 | |
| UCS | C1 ◁ C3 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2289 |
447 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27468, 167);