Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#11716033 |
| Sylow | C72 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #87641 |
| commutator, cosocle | C4×C72 | C23 |
| Frattini | C4 | C2×D72 |
| Fitting | C2×C4×C72 | C22 |
| socle | C22×C72 | D4 |
| hypercenter | C2×C4 | D72 |
| hypocenter | C72 | C2×D8 |
| LCS | G ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
233 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 233 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 |
164 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13096364, C2×#13096375, C2×#13096291, … |
| maximal normal subgroup of | C2×#13096364, C2×#13096375, C2×#13096291 |
| maximal quotient of | D3×#11716033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 456);