Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C4 |
| direct product | C7×C8×Dic7 |
| derived | C7 ⋊ C4×C7×C8 |
| Sylow | C72 ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7×C8 | D7 |
| commutator, hypocenter | C7 | C4×C7×C8 |
| Frattini | C2×C4 | C2×C7×D7 |
| Fitting | C2×C72×C8 | C2 |
| socle, prime radical | C22×C72 | C2×C4 |
| cosocle | C2×C4×C7 | C22×C7 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C7×C8 | |
121 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 256);