Finite Group DatabasebetaFinite Group Databasebeta| order | 27006 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×D643 |
| Fitting, socle, derived, Hall | C7×C643 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C643 ⋊ C2 |
| Hall | C7×D643 ⋊ C3 |
| Sylow | C643 ⋊ C2×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×D643 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C643 | C2×C3 |
| LCS | G ▷ C7×C643 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 4501 |
1615 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27006, 14);