Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C72 ⋊ #117 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D72 |
| commutator | C2×C72 | C22×C4 |
| Frattini | C22 | C2×D72 |
| Fitting | C2×C4×C72 | C22 |
| socle, cosocle | C22×C72 | C23 |
| prime radical | C22×C7×D7 | C22 |
| hypocenter | C72 | #117 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2×C4 | |
197 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^5,7^2|b^2=d,c^2=e,b^a=be,f^a=f^6,g^b=g^6188 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 424);