Finite Group DatabasebetaFinite Group Databasebeta| order | 644 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C23 · Schur multiplier (H2) = C2 |
| direct product | C2×D7×C23 |
| unambiguous semidirect product | C7⋊(C22×C23) |
| unambiguous semidirect product | (C2×C7)⋊(C2×C23) |
| Fitting, socle | C2×C7×C23 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C23 |
| Hall | C7×C23 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C23 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C23 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C23 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 322 |
230 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Dic23, #13046226, #13046228, … |
| maximal normal subgroup of | D7×Dic23, #13046226, #13046228, … |
| maximal quotient of | C23×Dic14, C4×D7×C23, C23×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(644, 7);