Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C72 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D72 |
| commutator, socle, cosocle | C22×C72 | C23 |
| Fitting | C23×C72 | C22 |
| prime radical | C2×C7×C22⋊D7 | C2 |
| hypercenter | C23 | D72 |
| hypocenter | C72 | #123 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C23 | |
237 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^5,7^2|b^2=d,b^a=be,c^a=cde,f^a=f^6,c^b=cde,g^b=g^6146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 562);