Finite Group DatabasebetaFinite Group Databasebeta| order | 7552 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C59×#536 ⋊ C2 |
| Sylow | C59 ⋊ #2009 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16837310 |
| commutator | C22×C4×C59 | C23 |
| Frattini | C22×C4 | C22×D59 |
| Fitting | C59×#536 | C2 |
| socle | C2×C59 | #392 |
| hypercenter | #536 | D59 |
| hypocenter | C59 | #2009 |
| LCS | G ▷ C22×C4×C59 ▷ C2×C59 ▷ C59 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #536 | |
348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,59|c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^58,c^b=cf,d^b=dg,d^c=dg,e^c=eg661 conjugacy classes — view the full character table.
| maximal subgroup of | D59×#2009, C2×#21425156, #28615696, … |
| maximal normal subgroup of | D59×#2009, C2×#21425156, #28615696, … |
| maximal quotient of | #28489004, #28489018, #28489032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7552, 5570);